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Counterexamples in Topology

Counterexamples in Topology is a foundational reference book in point-set topology, written by mathematicians Lynn Arthur Steen and J. Arthur Seebach Jr., and first published in 1970 by Holt, Rinehart and Winston.^[1] The work assembles 143 meticulously constructed counterexamples to illuminate the intricacies of topological concepts, disprove intuitive generalizations, and exemplify definitions, theorems, and proof techniques, making it an indispensable supplement for undergraduate and graduate courses in topology.^[2] A second edition appeared in 1978, followed by a widely accessible Dover reprint in 1995.^[1] The book's structure is designed to facilitate both learning and reference. Part I offers a concise review of essential definitions, including limit points, separation axioms (such as regularity, normality, and complete regularity), compactness (both global and local variants), connectedness, and metric spaces.^[1] Part II catalogs the 143 counterexamples, each treated comprehensively with descriptions of their construction, properties, and implications, spanning topologies like the finite discrete space, the Cantor set, and the Niemytzki tangent disc topology.^[2] Part III addresses metrization theory, exploring conjectures and additional counterexamples related to embedding spaces in metric structures. Part IV provides practical tools, including over 25 Venn diagrams and reference charts that summarize property implications (e.g., separation axiom hierarchies and compactness relations), enabling quick identification of examples with specific attributes.^[2] Inspired by the earlier Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted, the volume emphasizes an inductive approach to topology, where concrete examples drive understanding and reveal the limitations of abstract theorems. Prerequisites are modest—basic set theory and point-set topology—making it suitable for advanced undergraduates, though its depth supports graduate-level exploration. The Mathematical Association of America has recommended it for inclusion in undergraduate mathematics libraries, underscoring its enduring pedagogical value. With hundreds of citations in subsequent topological research, Counterexamples in Topology remains a standard resource for clarifying the nuanced distinctions that define the field.

Introduction

Purpose and Role in Topology

In topology, a counterexample is defined as a specific topological space that satisfies certain axioms or properties while failing others, thereby illustrating the boundaries and limitations of topological theorems and definitions. These examples serve to demonstrate that implications between properties are often one-directional rather than bidirectional, such as showing that compactness implies connectedness but not conversely through carefully constructed spaces that are connected yet non-compact.^[1] Counterexamples play a crucial role in disproving conjectures and testing the validity of proposed generalizations in point-set topology, where abstract axioms like those in separation properties can lead to subtle failures without concrete illustrations. By providing tangible spaces that violate expected behaviors, they clarify distinctions between related concepts, such as Hausdorff and regular spaces, preventing erroneous assumptions about universal applicability.^[1]^[3] In pedagogy, counterexamples are indispensable for fostering deeper comprehension among students, offering concrete visualizations of abstract ideas and encouraging creative engagement with the subject, much like the exploratory research that inspired seminal collections of such spaces. In research, they drive theoretical advancement by revealing gaps in existing proofs and stimulating new inquiries, underscoring that the pursuit of counterexamples is a vibrant aspect of mathematical discovery in topology.^[1]^[3] Moreover, counterexamples guard against overgeneralization from familiar settings like Euclidean spaces, where many topological properties coincide, by highlighting pathological behaviors in more general topological frameworks that might otherwise go unnoticed.^[1]

Historical Development

The development of counterexamples in topology began in the early 20th century with Felix Hausdorff's foundational work on separation axioms. In his 1914 monograph Grundzüge der Mengenlehre, Hausdorff introduced an axiomatic framework for topological spaces that included the T₂ separation axiom, now known as the Hausdorff axiom, which requires distinct points to be separable by disjoint open neighborhoods. This axiomatization highlighted the need for concrete examples to test the boundaries of these properties, paving the way for initial counterexamples illustrating failures in separation, such as non-Hausdorff spaces where points cannot be adequately separated.^[4] In the mid-1920s, Pavel Urysohn advanced the field significantly with his lemma, published posthumously in 1925, which characterizes normal spaces through the existence of continuous functions separating disjoint closed sets.^[5] This result, from Urysohn's paper "Mémoire sur les multiplicités Cantoriennes" in Fundamenta Mathematicae, not only clarified the implications of normality but also spurred the construction of counterexamples demonstrating that regularity does not imply normality, such as certain tangent disc spaces where closed sets fail to be separated despite the space being regular.^[6]^[7] These examples underscored the subtle distinctions in separation properties and influenced subsequent research into weaker and stronger axioms. Following World War II, the systematic compilation of counterexamples gained prominence with the 1970 publication of Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach, Jr., which assembled 143 illustrative examples across various topological concepts. The second edition in 1978 expanded this collection, incorporating additional cases and fostering a more rigorous, example-driven approach to topology education and research. This work marked a turning point, encouraging topologists to prioritize concrete pathologies to refine theorems and axioms. Post-1978 developments extended counterexample construction into specialized areas like paracompactness and dimension theory, with examples revealing failures in refinement properties and inductive dimensions, such as non-paracompact subspaces of paracompact spaces.^[8] In the 2020s, influences from digital topology introduced new counterexamples, particularly in topological complexity for digital images, where classical results fail due to discrete structures, as explored in studies adapting continuous theorems to pixelated settings.^[9]

Prerequisites and Notation

Basic Topological Concepts

A topological space consists of a set X together with a collection \tau of subsets of X, called open sets, satisfying three axioms: the empty set \emptyset and X itself belong to \tau; the union of any arbitrary collection of sets in \tau is in \tau; and the intersection of any finite collection of sets in \tau is in \tau.^[4] This structure generalizes familiar notions of continuity and proximity from metric spaces to more abstract settings without requiring a distance function.^[10] The complement of an open set in \tau is called a closed set, and a set is both open and closed (a clopen set) if it satisfies both properties.^[10] Closed sets satisfy dual axioms: the whole space and empty set are closed, arbitrary intersections of closed sets are closed, and finite unions of closed sets are closed.^[10] These definitions ensure that the topology is closed under the relevant operations, providing a foundation for studying limits and convergence in abstract spaces.^[10] A function f: X \to Y between topological spaces (X, \tau_X) and (Y, \tau_Y) is continuous if for every open set V \in \tau_Y, the preimage f^{-1}(V) is open in \tau_X.^[10] This preimage condition captures the intuitive idea that continuous functions preserve openness, allowing the extension of epsilon-delta continuity from metric spaces to general topologies.^[10] A base (or basis) for a topology \tau on X is a subcollection \mathcal{B} \subseteq \tau such that every open set in \tau can be expressed as a union of elements from \mathcal{B}.^[10] Bases simplify the description of topologies, as seen in the standard Euclidean topology where open balls form a base, and they are crucial for constructing subspaces and quotient spaces.^[10] For a subset A \subseteq X, the subspace topology on A is the collection \{U \cap A \mid U \in \tau\}, consisting of all intersections of open sets in X with A.^[10] This induced topology ensures that A inherits the "openness" properties relative to X, preserving continuity and other topological features within the subset.^[10] Separation axioms, which specify conditions for distinguishing points or sets via disjoint open neighborhoods, build upon these foundational concepts to classify spaces further.^[4]

Standard Notation and Terminology

In the study of counterexamples in topology, consistent notation facilitates precise discussion of pathological spaces and properties. A topological space is commonly denoted as (X, \tau), where X is the underlying set and \tau is the topology, consisting of the family of open subsets of X that satisfy the standard axioms (arbitrary unions and finite intersections closed under the collection, with \emptyset, X \in \tau). This notation emphasizes the structure imposed by \tau on X.^[11] Key set-theoretic operations in topological spaces employ conventional symbols to denote derived structures. The closure of a subset A \subseteq X, the smallest closed set containing A, is typically written as \mathrm{cl}(A) or \overline{A}. The interior of A, the largest open set contained in A, is denoted \mathrm{int}(A). The boundary of A is given by \mathrm{bd}(A) = \mathrm{cl}(A) \setminus \mathrm{int}(A), capturing points where A meets its complement in a limiting sense. These symbols appear routinely in analyses of counterexamples to distinguish limit behaviors and openness failures.^[11]^[12] Separation axioms, central to many counterexamples, use the T_n nomenclature originating from early 20th-century developments. The axiom T_0, also called the Kolmogorov axiom, requires that for distinct points x, y \in X, there exists an open set containing one but not the other. T_1 (Fréchet axiom) strengthens this so that each singleton \{x\} is closed, equivalent to distinct points having open neighborhoods excluding each other. T_2 (Hausdorff axiom) demands disjoint open neighborhoods for distinct points, ensuring unique limits for sequences. Higher axioms build on these: T_3 combines regularity (disjoint open sets separating a point from a disjoint closed set) with T_0 or T_1 (depending on convention), while T_4 pairs normality (disjoint open sets separating disjoint closed sets) with T_1. These terms reflect the progressive ability to "separate" points or sets via open covers.^[13]^[12] Historical variations in separation axiom terminology have led to inconsistencies across texts, complicating comparisons of counterexamples. Early definitions often omitted lower axioms; for instance, some pre-1950s works defined "regular" (old T_3) without T_0 and "normal" (old T_4) without T_1, allowing non-Hausdorff examples to satisfy them. Modern standardization, influenced by post-war surveys, typically requires T_3 = regular + T_1 and T_4 = normal + T_1 to align with metrizable spaces like \mathbb{R}^n. Russian literature occasionally reverses T_3 and T_4 labels, underscoring the need for explicit definitions in counterexample discussions.^[14]^[12] Topological properties relevant to counterexamples are sometimes abbreviated for brevity in diagrams and classifications. Compactness is denoted K, connectedness C, the Lindelöf property (every open cover has a countable subcover) L, and second-countability (countable basis for the topology) SC. These shorthand forms aid in cataloging spaces that fail implications, such as non-second-countable Lindelöf spaces.^[1] Literature on counterexamples exhibits minor notational differences for specific constructions, reflecting evolving emphases. The Sorgenfrey line, a classic non-normal hereditarily Lindelöf space, is the real line \mathbb{R} equipped with basis elements [a, b) for a < b, often symbolized as \mathbb{R}_s or simply S to highlight its lower-limit structure. In contrast, the Michael line, illustrating paracompactness without full normality in products, modifies \mathbb{R} by endowing rationals \mathbb{Q} with the standard topology and irrationals \mathbb{R} \setminus \mathbb{Q} with the discrete topology (open sets as standard opens union arbitrary subsets of irrationals); it is typically denoted M. Such variations, while not universal, appear in seminal collections to streamline references to these non-metrizable examples.^[1]^[15]

Counterexamples in Separation Properties

Non-Hausdorff and Weaker Spaces

In topology, separation axioms provide a hierarchy of conditions that ensure points or closed sets can be distinguished by open sets. The Kolmogorov axiom T0 requires that for any two distinct points, there exists an open set containing one but not the other. The Fréchet axiom T1 strengthens this by requiring that singletons are closed sets. The Hausdorff axiom T2 further demands that any two distinct points can be separated by disjoint open neighborhoods. Counterexamples illustrate that these axioms are independent, with spaces satisfying T0 but failing T1, and spaces satisfying T1 but failing T2.^[16] The Sierpiński space provides a minimal example of a T0 space that is not T1. Consider the set X = \{0, 1\} equipped with the topology \tau = \{\emptyset, \{1\}, X\}. This space is T0 because the open set \{1\} contains 1 but not 0, distinguishing the points. However, it fails T1 since the singleton \{1\} is not closed—its complement \{0\} is not open—while \{0\} is closed.^[16] The Sierpiński space also appears as Example 44 in Steen and Seebach's catalog of counterexamples, highlighting its role in demonstrating the distinction between T0 and T1.^[3] Another example of a T0 space that is not T1 is the particular point topology (also known as the included point topology) on an uncountable set X with a distinguished point p \in X. The open sets are \emptyset and all subsets of X containing p. For distinct points q \neq r in X, if neither is p, then X \setminus \{q\} is open and contains p but not q; if one is p, say q = p, then any open set containing r (which must include p) also contains q, but the open set \{p\} contains q but not r. Thus, the space is T0. It fails T1 because the singleton \{p\} is not closed—its complement X \setminus \{p\} is not open—while singletons \{q\} for q \neq p are closed. This topology is particularly useful for uncountable sets to emphasize the special role of the point p.^[3] The cofinite topology on an infinite set X exemplifies a T1 space that is not Hausdorff. The open sets are \emptyset and all subsets with finite complements in X, so closed sets are X and all finite subsets. This satisfies T1 because every singleton \{x\} is finite, hence closed, ensuring complements are open. However, it fails T2: for distinct points x, y \in X, any nonempty open sets U, V must each omit only finitely many points, so their intersection omits at most finitely many and is nonempty since X is infinite, preventing disjoint neighborhoods. This example, known as the finite complement topology, is the coarsest T1 topology on an infinite set and appears as Example 19 in Steen and Seebach.^[16]^[3] These counterexamples underscore key implications in separation theory: T0 does not imply T1, as seen in the Sierpiński and particular point topologies where points cannot be individually closed despite distinguishability; moreover, T1 does not imply T2, as the cofinite topology shows non-separated points via unavoidable open set intersections. Such spaces are non-Hausdorff overall, revealing that weaker axioms permit pathological behaviors like non-closed singletons or inseparable points, which stronger axioms like regularity (detailed elsewhere) aim to exclude.^[16]^[3]

Failures in Regularity and Normality

In topology, a space is defined as regular if it is T_0 (Kolmogorov) and, for every point x and closed set C with x \notin C, there exist disjoint open sets U containing x and V containing C.^[17] This property strengthens the Hausdorff condition by ensuring points can be separated from closed sets not containing them. A space is normal if it is T_1 (Fréchet) and, for any two disjoint closed sets A and B, there exist disjoint open sets U containing A and V containing B.^[18] Normality extends regularity to the separation of arbitrary disjoint closed sets and implies complete regularity, where points and closed sets can be separated by continuous functions to [0,1]. While every normal space is regular, the converse fails, as demonstrated by various counterexamples in Hausdorff spaces. These examples illustrate breakdowns in higher separation axioms, often preserving regularity or complete regularity but failing normality due to structural limitations in the topology. Seminal constructions, such as those involving modified Euclidean spaces or ordinal products, highlight how local separation can coexist with global inseparability of closed sets. The Niemytzki plane, also known as the Moore plane, provides a classic instance of a Hausdorff regular space that is not normal (Example 6 in Steen and Seebach). Constructed on the set X = P \cup L, where P is the open upper half-plane \{(x,y) \in \mathbb{R}^2 \mid y > 0\} and L is the x-axis \mathbb{R} \times \{0\}, the topology uses Euclidean open disks as a basis for points in P, while basis elements at points x \in L are \{x\} \cup D, with D an open disk in P tangent to L at x and having rational radius.^[3] This space is Hausdorff and regular, as points can be separated using the tangent disk neighborhoods, and it is also developable (a Moore space) and locally compact. However, it fails normality because every subset of L is closed, so the rationals Q \subset L and irrationals I \subset L form disjoint closed sets with no disjoint open neighborhoods: any open set around a rational on L intersects every neighborhood of nearby irrationals due to the dense rational radii.^[3] Another prominent counterexample is the deleted Tychonoff plank (Example 105 in Steen and Seebach), which is completely regular but not normal. This space is the subspace T^* = ([0, \omega_1] \times [0, \omega]) \setminus \{(\omega_1, \omega)\} of the product of ordinal spaces, where \omega_1 is the first uncountable ordinal and \omega is the first infinite ordinal, equipped with the order topology.^[3] As a subspace of the compact Hausdorff Tychonoff plank, T^* inherits complete regularity and is also locally compact. Yet, it is not normal, as the disjoint closed sets A = \{(\omega_1, n) \mid 0 \leq n < \omega\} and B = \{(a, \omega) \mid 0 \leq a < \omega_1\} cannot be separated by disjoint open sets: any neighborhood of A must include points near (\omega_1, \omega), which forces intersection with neighborhoods of B due to the deleted corner point.^[3] Uncountable products of discrete spaces further exemplify regular spaces that fail normality (Example 103 in Steen and Seebach). Consider the product X^\Lambda = \prod_{\lambda \in \Lambda} \mathbb{N}, where \Lambda is uncountable and \mathbb{N} = \{0,1,2,\dots\} carries the discrete topology.^[3] This space is completely regular (as a product of Tychonoff spaces) and separable when |\Lambda| \leq 2^{\aleph_0}. However, it is not normal: certain disjoint closed sets, such as specific uncountable closed discrete subsets, cannot be separated by disjoint open sets due to the uncountable index set allowing coordinates to vary in ways that force intersections in any attempted open covers.^[3] This construction underscores how product topologies can preserve point-set separation while collapsing under uncountable complexity.

Counterexamples in Compactness and Local Properties

Non-Compact Locally Compact Spaces

A topological space X is locally compact if for every point x \in X, there exists a compact neighborhood of x, or equivalently, a neighborhood basis at x consisting of compact sets.^[19] In contrast, X is compact if the entire space X is a compact set, meaning every open cover has a finite subcover. Local compactness implies that the space behaves like a compact space "locally" at each point, but it does not guarantee global compactness, leading to numerous counterexamples that illustrate the distinction in topological properties. The real line \mathbb{R} with its standard topology serves as a fundamental example of a locally compact but non-compact space. Every point x \in \mathbb{R} has a compact neighborhood, such as the closed interval [x-1, x+1], which is compact by the Heine-Borel theorem.^[20] However, \mathbb{R} itself is not compact, as the open cover \{(-n, n) \mid n \in \mathbb{N}\} has no finite subcover.^[20] A more advanced counterexample is the long ray, defined as the set [0, \omega_1) where \omega_1 is the least uncountable ordinal, equipped with the order topology. The long ray is locally compact, as it is locally homeomorphic to the real line \mathbb{R} (which is locally compact) at every point except the origin, where [0, \omega) provides a compact neighborhood.^[21] Nonetheless, the long ray is not compact, as the collection of open intervals [0, \alpha) for all \alpha < \omega_1 forms an open cover with no finite subcover; moreover, it requires uncountably many such sets, so the space is not \sigma-compact.^[21] The long ray is connected and even first countable but highlights how ordinal topologies can extend familiar properties like local compactness indefinitely without achieving compactness. Local compactness also plays a crucial role in the construction of compactifications. The Alexandroff one-point compactification of a non-compact Hausdorff space X, denoted \alpha X = X \cup \{\infty\}, adds a single point at infinity and defines open sets as either open subsets of X or complements in \alpha X of compact subsets of X.^[22] This construction yields a compact space, but \alpha X is Hausdorff if and only if X is locally compact.^[23] To see the failure without local compactness, consider X = \mathbb{Q} with the subspace topology inherited from \mathbb{R}. Here, \mathbb{Q} is Hausdorff but not locally compact, since compact subsets of \mathbb{Q} are finite and no point admits a compact infinite neighborhood.^[20] In \alpha \mathbb{Q}, the point \infty cannot be separated from any q \in \mathbb{Q} by disjoint open sets: any neighborhood of \infty is the complement of a finite set in \mathbb{Q} (hence cofinite), while any neighborhood of q is infinite and thus intersects it. Consequently, \alpha \mathbb{Q} is compact but fails to be Hausdorff.^[24]

Paracompactness and Collectionwise Norms

A topological space is paracompact if every open cover admits a locally finite open refinement.^[1] This property strengthens compactness in certain ways, ensuring refinements that are both open and locally finite, which aids in embedding spaces and extending continuous functions. Paracompactness is particularly useful in manifold theory and uniformization, but counterexamples reveal its subtleties, especially in products and separation properties.^[1] The Sorgenfrey line provides a key counterexample illustrating limitations of paracompactness under products. Defined as the real line \mathbb{R} equipped with the topology generated by half-open intervals [a, b) for a < b, it is hereditarily Lindelöf and thus paracompact. However, its Cartesian square, the Sorgenfrey plane, fails to be normal: the set K = \{(x, -x) \mid x \in \mathbb{Q}\} of rational points on the anti-diagonal and the set L = \{(x, -x) \mid x \in \mathbb{R} \setminus \mathbb{Q}\} of irrational points on the anti-diagonal cannot be separated by disjoint open sets, as any basic open neighborhoods in the Sorgenfrey topology intersect.^[1] This shows that paracompactness does not preserve normality in products, despite the line being perfectly normal. Collectionwise normality refines the notion of normality by requiring that every discrete collection of closed sets can be separated by pairwise disjoint open sets.^[1] This property is crucial for metrization theorems and embedding results, but counterexamples demonstrate that it does not follow from weaker separation axioms or even from connectivity. The Knaster–Kuratowski fan, constructed in the plane as the union of line segments joining the apex (0, 1/2) to points in the Cantor set on the x-axis—specifically, including all heights over irrational Cantor points and only rational heights over endpoint Cantor points—serves as a connected space that is not collectionwise normal. The rational endpoints form an infinite closed discrete subset, but no collection of pairwise disjoint open sets can separate them, as any neighborhood of an endpoint intersects infinitely many irrational spines, forcing overlaps near the apex.^[1] Even normality does not imply collectionwise normality, as shown by Bing's Example G: a subspace of the plane constructed by attaching shrinking wedges of circles to rational points on the y-axis in a way that ensures overall normality but fails collectionwise separation for the attached closed sets. This space is normal, countably paracompact, and first countable, yet the discrete collection of the bases of the wedges cannot be separated disjointly without intersecting due to the dense attachment.^[1] Such examples underscore the gap between pointwise and collectionwise separation, impacting results like the Bing metrization theorem, which requires collectionwise normality for Moore spaces.

Counterexamples in Connectedness and Path Properties

Connected but Not Path-Connected Spaces

In topology, a space X is connected if it cannot be expressed as the union of two nonempty disjoint open subsets, meaning there is no nontrivial clopen separation of X.^[25] A stronger property is path-connectedness, where for any two points x, y \in X, there exists a continuous function \gamma: [0,1] \to X such that \gamma(0) = x and \gamma(1) = y. Path-connectedness implies connectedness, but the converse does not hold, as demonstrated by several classic counterexamples.^[25] One of the most famous examples is the topologist's sine curve, defined as the subspace S = \{(x, \sin(1/x)) \mid 0 < x \leq 1\} \cup \{(0, y) \mid -1 \leq y \leq 1\} of \mathbb{R}^2 equipped with the standard topology (Example 116 in Steen and Seebach). This space is connected because any open neighborhood of a point on the vertical segment \{0\} \times [-1,1] must intersect the oscillating curve \{(x, \sin(1/x)) \mid 0 < x \leq 1\}, preventing a separation into disjoint open sets. However, S is not path-connected: there is no continuous path connecting a point on the vertical segment, such as (0,0), to a point on the curve, such as (1, \sin(1)). Any purported path \gamma: [0,1] \to S from (0,0) to (1, \sin(1)) would require the image \gamma^{-1}(\{0\} \times [-1,1]) to be compact and connected in [0,1], hence an interval, but the infinite oscillations of the sine curve force this preimage to have uncountably many components, leading to a contradiction. A related counterexample is the Warsaw circle (also known as the topologist's sine circle), constructed by attaching an arc to the topologist's sine curve to close it into a loop. Specifically, it consists of the oscillating curve \{(x, \sin(1/x)) \mid 0 < x \leq 1\}, the vertical segment \{0\} \times [-1,1], and a semicircular arc connecting (1, \sin(1)) to (0,1), all in the subspace topology of \mathbb{R}^2. This space is connected for similar reasons as the topologist's sine curve, as the attachments prevent separation. Moreover, the Warsaw circle is path-connected, as the arc allows continuous paths between any two points by going around the loop. However, it is not locally path-connected: neighborhoods of points on the vertical segment near the attachment include parts of the infinitely oscillating curve that form disconnected quasi-components, preventing a basis of path-connected open sets. This example highlights that path-connectedness does not imply local path-connectedness in connected spaces.^[26] These counterexamples illustrate the limitations of connectedness as a topological invariant, showing that paths may fail to exist between points in otherwise "intact" spaces, which has implications for studying homotopy and fundamental groups in algebraic topology. The topologist's sine curve and Warsaw circle are seminal in demonstrating how pathological behaviors arise from simple modifications to familiar objects like curves and circles. A topological space X is said to be locally connected if it has a basis consisting of connected open subsets. This property ensures that connectedness holds locally at every point, meaning every point has arbitrarily small connected neighborhoods. While local connectedness is a strengthening of local properties in connected spaces, connectedness does not imply local connectedness, leading to significant pathologies in the structure of connected components. In such spaces, connected components may not be open, resulting in points that are inaccessible within their components, where no connected open neighborhood is contained entirely in the component. This failure has implications for further properties like path-connectedness and the behavior of continuous functions. One classic counterexample is the topologist's comb space, defined as the subset C = ([0,1] \times \{0\}) \cup \bigcup_{n=1}^\infty \{\frac{1}{n}\} \times [0,1] \cup \{0\} \times [0,1] of \mathbb{R}^2 equipped with the subspace topology (Example 107 in Steen and Seebach). This space is connected and even path-connected, as any two points can be joined by a path along the "teeth" and spine, but it is not locally connected at points on the spine \{0\} \times (0,1], where small neighborhoods contain disconnected "hairs" that prevent a connected basis. The Knaster-Kuratowski connected fan provides another illustration of these implications (Example 129 in Steen and Seebach). Constructed from the unit disk by taking line segments from the origin (the apex) to points on the boundary Cantor set, with a modified basis at the apex that groups endpoints with rational ternary digits into sectors while isolating those with irrational digits to single rays, the space is connected but becomes totally disconnected upon removal of the apex. It is not path-connected, and locally connected at all points except the apex, where neighborhoods fail to be connected due to the dispersion of the "irrational" rays, making the apex an inaccessible point whose removal destroys all connectivity. This highlights how the absence of local connectedness at a single point can render the global connectedness fragile. The irrational slope topology on the rational upper half-plane X = \{(x,y) \in \mathbb{Q}^2 \mid y \geq 0\} further exemplifies a connected space lacking local connectedness (Example 75 in Steen and Seebach). The basis consists of "crosses" centered at rational points, formed by horizontal and vertical segments, augmented by strips of irrational slope \theta extending to small rational balls; this generates a connected, countable Hausdorff space that is not path-connected, but open sets around points are not connected, as the irrational strips intersect the rational grid in disconnected ways, preventing a connected local basis. Such constructions demonstrate implications for accessibility, where points in these spaces cannot be approached through connected paths within small neighborhoods, affecting properties like the continuity of retractions and embeddings. Conversely, local connectedness does not imply local path-connectedness, showing further separation in path properties. The lexicographically ordered unit square [0,1] \times [0,1], with the order topology generated by the lexicographic order (first by x-coordinate, then y), is connected and locally connected, as basis elements are order-intervals that are connected (Example 82 in Steen and Seebach). However, it is not locally path-connected, since neighborhoods of points like (0,0) contain vertical fibers that are totally ordered but lack paths connecting points across different x-slices due to the rigid order, violating the requirement for path-connected local bases. This pathology underscores that while local connectedness ensures local topological integrity, it does not guarantee local path accessibility, impacting applications in algebraic topology such as fundamental group computations.

Counterexamples in Metrizability and Uniformity

Non-Metrizable Topological Spaces

A topological space is metrizable if its topology can be induced by a metric. While many familiar spaces like the real line are metrizable, there exist spaces that satisfy several properties associated with metric spaces—such as being Hausdorff, regular, first-countable, or paracompact—but still fail to be metrizable. The Nagata–Smirnov metrization theorem provides a complete characterization: a topological space X is metrizable if and only if it is regular Hausdorff and possesses a basis that is \sigma-locally finite, meaning the basis is a countable union of locally finite collections of open sets.^[27] This condition generalizes earlier results, as a second-countable space (one with a countable basis) admits a \sigma-locally finite basis, since the countable basis itself is a countable union of singleton families, each of which is locally finite. Consequently, the theorem implies Urysohn's metrization theorem, which states that every regular second-countable Hausdorff space is metrizable.^[28] The long line serves as a prominent counterexample illustrating the necessity of the \sigma-locally finite basis condition. Constructed as the lexicographic order topology on [0, \omega_1) \times [0,1) where \omega_1 is the first uncountable ordinal, the long line is Hausdorff, first-countable (hence regular), paracompact, connected, locally compact, and sequentially compact.^[21] However, it lacks a \sigma-locally finite basis and is not metrizable. To see this directly, note that the long line is not compact: the cover by intervals [0, \alpha) \times [0,1) for successor ordinals \alpha < \omega_1 has no finite subcover. Yet it is sequentially compact, as any sequence lies within a countable ordinal segment, which is compact. In contrast, every sequentially compact metric space is compact, so the long line cannot admit a compatible metric.^[29] This example demonstrates that first-countability and paracompactness, while strengthening regularity, are insufficient for metrizability without the basis condition. In contrast, the uncountable discrete space—formed by equipping an uncountable set with the discrete topology—highlights subtleties in the necessity of second-countability. This space is first-countable (with singletons as a local basis at each point), regular Hausdorff, and metrizable via the discrete metric d(x,y) = 1 if x \neq y and d(x,x) = 0. However, it is not second-countable, as any basis must include all singletons, forming an uncountable collection. Its \sigma-locally finite basis consists of the singletons themselves, which form a single locally finite family (since neighborhoods intersect at most one singleton). This shows that second-countability is not necessary for metrizability in general Hausdorff spaces; rather, in metrizable spaces, second-countability is equivalent to separability, as non-separable metric spaces like the uncountable discrete space exist but lack countable dense subsets. The long line and such examples underscore that the conditions in metrization theorems are sharp: weakening them, as in the absence of a \sigma-locally finite basis, can yield non-metrizable spaces despite other metric-like properties.^[30]

Uniform Spaces Without Metrizable Uniformity

A uniform space consists of a set X together with a filter \mathcal{U} on X \times X (the entourages) such that \Delta_X \subseteq \bigcap_{U \in \mathcal{U}} U, if U \in \mathcal{U} then U^{-1} \in \mathcal{U}, for each U \in \mathcal{U} there exists V \in \mathcal{U} such that V \circ V \subseteq U, and \mathcal{U} is upward closed (if U \in \mathcal{U} and U \subseteq W then W \in \mathcal{U}).^[31] This structure induces a topology on X where a set V \subseteq X is open if for every x \in V, there exists U \in \mathcal{U} such that V(x) = \{y \in X \mid (x,y) \in U\} \subseteq V. A uniformity \mathcal{U} on X is metrizable if there exists a metric d on X such that the entourages induced by d—namely, the sets \{(x,y) \in X \times X \mid d(x,y) < \epsilon\} for \epsilon > 0—form a base for \mathcal{U}. Equivalently, \mathcal{U} is metrizable if and only if it admits a countable base consisting of symmetric entourages. A prominent counterexample arises in the context of infinite products, as detailed in Counterexamples in Topology (Example 44). Consider the uncountable discrete ordinal space: an uncountable set X of limit ordinals with the discrete topology, equipped with a non-metrizable uniformity U_2 whose base consists of entourages B_z = \Delta \cup \{(x, y) \mid x > z, y > z\} for ordinals z. This uniformity induces the discrete (metrizable) topology but lacks a countable base due to the uncountable index set, hence is non-metrizable.^[3] Similarly, for an uncountable index set A and, for each \alpha \in A, a nontrivial metric space (X_\alpha, d_\alpha) with at least two points (e.g., the two-point discrete space). The product space \prod_{\alpha \in A} X_\alpha can be equipped with the product uniformity \mathcal{U}, defined as the initial uniformity making all projection maps \pi_\alpha: \prod_{\beta \in A} X_\beta \to X_\alpha uniformly continuous. A base for \mathcal{U} consists of entourages of the form \prod_{\alpha \in A} U_\alpha, where U_\alpha \in \mathcal{U}_{d_\alpha} (the uniformity induced by d_\alpha) for \alpha in some finite subset F \subseteq A, and U_\alpha = X_\alpha \times X_\alpha otherwise. The topology induced by \mathcal{U} is the product topology, which is not metrizable when A is uncountable, as it fails to be second-countable (e.g., the standard basis elements require specifying behavior in uncountably many coordinates). Consequently, since a metrizable uniformity would induce a metrizable topology, \mathcal{U} cannot be metrizable.^[32] Bing's metrization theorem characterizes metrizable spaces as regular Hausdorff with a \sigma-discrete basis, and his discrete extension space provides a counterexample of a uniformizable but non-metrizable Hausdorff space.^[33] These examples underscore the nuanced role of uniformity in extending metric concepts. The implications of non-metrizable uniformities extend to analysis and topology, where such structures allow for quasi-metrics or coarser uniformities that capture uniform continuity without full metric completeness. In infinite products, this failure enables the study of phenomena like uniform convergence in function spaces that depend on uncountably many variables, without the restrictive countable basis of metrics.

Notable Specific Counterexamples

The Long Line and Variants

The long line, also known as the Alexandroff line, is constructed by taking the first uncountable ordinal \omega_1 and forming the set \omega_1 \times [0, 1) ordered lexicographically, where (\alpha, t) < (\beta, s) if \alpha < \beta or if \alpha = \beta and t < s. The topology is the order topology induced by this total order, making it a linearly ordered topological space (LOTS). Equivalently, it can be viewed as adjoining uncountably many copies of the half-open interval [0, 1) in sequence, one for each countable ordinal, with the order topology.^[21]^[34] This space is locally Euclidean: every point has an open neighborhood homeomorphic to the real line \mathbb{R}, as basic open intervals around points mimic those in \mathbb{R}. It is connected and path-connected, with any two points joined by a continuous path homeomorphic to an interval in \mathbb{R}. The long line is Hausdorff, regular, completely regular, normal, first countable, countably compact, and completely normal due to its order topology. However, it fails to be second-countable, as any countable basis would contradict the uncountability of \omega_1; it is also not separable, \sigma-compact, Lindelöf, metrizable, or paracompact, with the latter failure arising because no locally finite open cover refines the uncountable disjoint collection of "segments" corresponding to successor ordinals. Limit ordinals form closed sets that are not G_\delta, so it is not perfectly normal.^[21]^[34]^[35] A key variant is the extended long line, obtained by adjoining a point at "infinity" to the long line, yielding the compact space [0, \omega_1] in the order topology. This addition makes the space compact and connected but not path-connected, as no continuous path reaches the endpoint \omega_1, and it fails local Euclideanness at that point. Another variant, the long plane, is the topological product of the long line with the closed unit interval [0, 1], resulting in a non-paracompact 2-manifold. This construction demonstrates failures in dimension theory for non-metrizable spaces, where inductive and covering dimensions may not coincide or behave additively as in Euclidean cases, highlighting limitations of classical theorems like those of Urysohn or Hurewicz.^[21]^[34]^[36] The foundational ordinals underlying the long line were introduced by Georg Cantor in the late 19th century, but the space itself was constructed by Pavel Alexandrov in the 1920s as part of early investigations into ordered continua. It gained prominence as a counterexample through its inclusion in Steen and Seebach's 1970 catalog of topological pathologies.^[21]^[34]

Sorgenfrey Plane and Line

The Sorgenfrey line, denoted \mathbb{R}_\ell, is the set of real numbers \mathbb{R} equipped with the lower limit topology, generated by the basis of all half-open intervals [a, b) where a, b \in \mathbb{R} and a < b. This basis consists of sets that are open in the Sorgenfrey topology, making it strictly finer than the standard Euclidean topology on \mathbb{R}. The space \mathbb{R}_\ell is first countable, as each point x has a local basis given by the countable collection \{[x, x + 1/n) \mid n \in \mathbb{N}\}, but it is not second countable, since the uncountably many disjoint open sets [r, r+1) for each real r require an uncountable basis to cover them separately.^[37] The Sorgenfrey line possesses several notable properties that highlight its role as a counterexample in general topology. It is hereditarily Lindelöf, meaning every subspace is Lindelöf (every open cover has a countable subcover), and it is a Baire space, where the intersection of countably many dense open sets is dense. Additionally, \mathbb{R}_\ell is paracompact, admitting a locally finite open refinement for every open cover. These attributes demonstrate that \mathbb{R}_\ell satisfies many desirable covering and completeness properties despite lacking second countability.^[38]^[39] The Sorgenfrey plane, denoted \mathbb{R}_\ell \times \mathbb{R}_\ell, is the topological product of the Sorgenfrey line with itself, with basis elements of the form [a, b) \times [c, d). While each factor space is paracompact (and hence normal), the product fails to be normal, illustrating the limitations of product theorems in topology: paracompactness does not preserve under finite products in general. To see the failure of normality, consider the anti-diagonal D = \{(x, -x) \mid x \in \mathbb{R}\}, which is closed in the Sorgenfrey plane. The subsets A = \{(q, -q) \mid q \in \mathbb{Q}\} and B = \{(r, -r) \mid r \in \mathbb{R} \setminus \mathbb{Q}\} are disjoint and closed in the plane, but they cannot be separated by disjoint open sets; any open neighborhood of B must intersect A due to the density of the rationals and the structure of the basis elements. This construction, analogous to Sorgenfrey's original argument using rational and irrational points on a line segment, underscores how the half-open basis disrupts separation in the product.^[39]

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### Summary of Space S and S × S from the Document