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Euclidean topology

Euclidean topology, also known as the standard, usual, or classical topology on the Euclidean space \mathbb{R}^n, is the topology generated by the Euclidean metric, where the basic open sets are the open balls B(x, r) = \{ y \in \mathbb{R}^n \mid d(x, y) < r \} for points x \in \mathbb{R}^n and radii r > 0, with the Euclidean distance d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}.^[1] This topology makes \mathbb{R}^n a metric space, where open sets are arbitrary unions of these open balls, and it is independent of the choice of any equivalent norm on the finite-dimensional vector space \mathbb{R}^n.^[2] All norms on \mathbb{R}^n induce the same topology, ensuring that topological properties remain consistent across different metrics that generate it.^[2] Key properties of the Euclidean topology include being Hausdorff, meaning distinct points can be separated by disjoint open neighborhoods, which facilitates unique limits and convergence.^[1] It is also second-countable, possessing a countable basis of open sets (such as open balls centered at rational points with rational radii), making \mathbb{R}^n separable and Lindelöf.^[1] Additionally, the space is locally compact, as every point has a compact neighborhood (e.g., a closed ball of finite radius), and metrizable by the Euclidean metric, implying it is normal and paracompact.^[1] These attributes underpin its role as the foundational topological structure in real analysis, differential geometry, and manifold theory, where \mathbb{R}^n serves as a model for locally Euclidean spaces.^[3] The Euclidean topology extends naturally from the one-dimensional case on \mathbb{R}, where open sets are unions of open intervals, to higher dimensions via the product topology, preserving continuity of functions like projections and ensuring compatibility with multivariable calculus concepts such as limits and continuity.^[4] Closed sets in this topology are complements of open sets, including closed balls and their finite unions, while the space itself is connected and path-connected, allowing for the study of continuous deformations without tearing or gluing.^[1] This topology contrasts with coarser or finer ones but remains the canonical choice for embedding Euclidean spaces into broader topological frameworks.^[4]

Definition and Construction

The Euclidean Metric

The Euclidean metric on the n-dimensional real vector space \mathbb{R}^n is defined by the distance function

d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2},

where x = (x_1, \dots, x_n) and y = (y_1, \dots, y_n) are points in \mathbb{R}^n.^[5] This metric arises directly from the \ell_2 norm of the difference vector x - y, given by \|x - y\|_2 = \sqrt{(x - y) \cdot (x - y)}, where \cdot denotes the standard Euclidean inner product on \mathbb{R}^n. The formula \sqrt{\sum (x_i - y_i)^2} follows from the Pythagorean theorem applied to coordinate differences.^[5] The Euclidean metric derives its name from the geometric framework established by Euclid in his Elements around 300 BCE, where distances were defined via straight-line segments in the plane and space without numerical representation.^[6] The distance formula is based on the Pythagorean theorem and was introduced in coordinate geometry by René Descartes in the 17th century. Its generalization to norms in higher-dimensional \mathbb{R}^n (n > 3) appeared in the 19th century. The Euclidean metric satisfies the axioms of a metric space. It is non-negative, with d(x, y) \geq 0 for all x, y \in \mathbb{R}^n, and positive definite, meaning d(x, y) = 0 if and only if x = y; these follow from the non-negativity of squares and the injectivity of the norm.^[5] It is symmetric, so d(x, y) = d(y, x) for all x, y \in \mathbb{R}^n, which holds by the commutativity of subtraction and addition in the sum.^[5] The triangle inequality states that d(x, z) \leq d(x, y) + d(y, z) for all x, y, z \in \mathbb{R}^n; a proof sketch uses the Cauchy-Schwarz inequality |\langle u, v \rangle| \leq \|u\|_2 \|v\|_2 applied to u = x - y and v = y - z, yielding

\|x - z\|_2^2 = \|u + v\|_2^2 = \|u\|_2^2 + \|v\|_2^2 + 2 \langle u, v \rangle \leq \|u\|_2^2 + \|v\|_2^2 + 2 \|u\|_2 \|v\|_2 = (\|u\|_2 + \|v\|_2)^2,

and taking square roots gives the result.^[5] Although the Euclidean metric extends to infinite-dimensional \ell_2 spaces, forming complete inner product spaces known as Hilbert spaces, the standard Euclidean topology concerns only finite-dimensional \mathbb{R}^n.^[7]

Topology Induced by the Metric

The topology induced by the Euclidean metric d on \mathbb{R}^n is the coarsest topology \mathcal{T}_d on \mathbb{R}^n such that the metric d: \mathbb{R}^n \times \mathbb{R}^n \to [0, \infty) is continuous with respect to the product topology on \mathbb{R}^n \times \mathbb{R}^n and the standard topology on [0, \infty).^[8] This topology consists precisely of all arbitrary unions of open balls B_r(x) = \{ y \in \mathbb{R}^n \mid d(x, y) < r \} for x \in \mathbb{R}^n and r > 0.^[8] A set U \subseteq \mathbb{R}^n belongs to \mathcal{T}_d if and only if for every x \in U, there exists r > 0 such that B_r(x) \subseteq U.^[9] The collection \mathcal{B} = \{ B_r(x) \mid x \in \mathbb{R}^n, r > 0 \} of all open balls forms a basis for \mathcal{T}_d. To verify this, first note that \mathcal{B} covers \mathbb{R}^n since B_r(x) \ni x for any x and r > 0. For the basis condition, consider x \in B_{r_1}(x_1) \cap B_{r_2}(x_2) for some x_1, x_2 \in \mathbb{R}^n and r_1, r_2 > 0; let r = \min\{ r_1 - d(x, x_1), r_2 - d(x, x_2) \}, which is positive since d(x, x_1) < r_1 and d(x, x_2) < r_2. Then B_r(x) \subseteq B_{r_1}(x_1) \cap B_{r_2}(x_2). Thus, every open set in \mathcal{T}_d is a union of elements of \mathcal{B}.^[9] All norms on the finite-dimensional space \mathbb{R}^n are equivalent, in the sense that for any two norms \|\cdot\|_a and \|\cdot\|_b, there exist constants $0 < c \leq C < \infty such that c \|x\|_b \leq \|x\|_a \leq C \|x\|_b for all x \in \mathbb{R}^n.^[10] This equivalence implies that any two norms induce the same topology on \mathbb{R}^n, as the open balls in one norm are contained in scalar multiples of open balls in the other, and vice versa. In particular, the L_p norms \|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^{1/p} for $1 \leq p < \infty and \|x\|_\infty = \max_{1 \leq i \leq n} |x_i| all generate the same Euclidean topology \mathcal{T}_d, since each is equivalent to the Euclidean norm \| \cdot \|_2.^[10] The Euclidean topology \mathcal{T}_d is the unique Hausdorff topology on \mathbb{R}^n that makes it a topological vector space over \mathbb{R}, meaning addition \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n and scalar multiplication \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n are continuous with respect to the product topology on the domain.^[11] Further details on this characterization as a locally convex topological vector space are addressed in subsequent sections on structural properties.

Fundamental Properties

Separation Axioms

The Euclidean topology on \mathbb{R}^n satisfies the full spectrum of separation axioms, beginning with the Hausdorff condition and extending to complete normality, due to its origins as a metric space induced by the Euclidean metric d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}.^[12] These properties ensure that points and closed sets can be rigorously separated by open neighborhoods, distinguishing the Euclidean topology from coarser structures.^[13] The space is Hausdorff (T2), meaning that for any two distinct points x, y \in \mathbb{R}^n with x \neq y, there exist disjoint open sets containing each. Specifically, the open balls B_{d(x,y)/2}(x) and B_{d(x,y)/2}(y) serve as such neighborhoods, since their radii ensure no overlap given the triangle inequality in the metric.^[12] This separation arises directly from the positive distance between distinct points in the metric space.^[13] Euclidean space satisfies stronger separation axioms as well. It is regular (T3), so for any point x \in \mathbb{R}^n and closed set A with x \notin A, there exist disjoint open sets U \ni x and V \supset A; one constructs U as the open ball B_{\delta/2}(x) where \delta = \inf\{d(x,a) \mid a \in A\} > 0, and V as the union of balls around points of A with radius \delta/2.^[13] Similarly, it is normal (T4), allowing separation of any two disjoint closed sets A and B by disjoint open sets; when d(A,B) = \inf\{d(a,b) \mid a \in A, b \in B\} > 0, open "tubes" of radius d(A,B)/2 around each set provide the separation without overlap, while in general (including when d(A,B) = 0), separation is possible via the properties of metric spaces, such as Urysohn's lemma.^[13] The space is also completely normal (T5), as every subspace inherits normality from the metric structure.^[14] By construction, the Euclidean topology is metrizable via the given metric, which is complete: every Cauchy sequence converges to a point in \mathbb{R}^n.^[12] In contrast to the indiscrete topology, where the only open sets are the empty set and the whole space—preventing any separation of points—the Euclidean topology robustly distinguishes all distinct elements.^[15]

Countability Axioms

Euclidean spaces satisfy the first axiom of countability, meaning that for every point x \in \mathbb{R}^n, there exists a countable local basis at x. A standard such basis consists of the open balls B(x, 1/n) for n = 1, 2, \dots, which shrink towards x and generate all neighborhoods of x due to the metric structure.^[16] This property holds generally for all metric spaces, including the Euclidean metric on \mathbb{R}^n.^[17] The stronger second axiom of countability also holds for \mathbb{R}^n, where the space admits a countable basis for its topology. One such basis is formed by all open balls B(q, r) where q \in \mathbb{Q}^n and r \in \mathbb{Q}^+, the set of positive rational numbers; the countability arises from the fact that both \mathbb{Q}^n and \mathbb{Q}^+ are countable, and their product yields a countable collection that generates the entire topology via unions.^[18] Equivalently, in the product topology, the basis of open rectangles with rational endpoints provides another countable basis.^[19] As a consequence of second countability, \mathbb{R}^n is separable: it contains a countable dense subset, namely \mathbb{Q}^n. To see the density, for any x \in \mathbb{R}^n and \epsilon > 0, there exist rationals q_1, \dots, q_n \in \mathbb{Q} such that |x_i - q_i| < \epsilon / \sqrt{n} for each coordinate i, ensuring \|x - q\| < \epsilon by the Euclidean norm, leveraging the continuity of the distance function and the density of \mathbb{Q} in \mathbb{R}.^[20]^[21] Second countability further implies that \mathbb{R}^n is paracompact, as it is a second-countable, locally compact Hausdorff space; paracompactness ensures that every open cover admits a locally finite open refinement, which in turn allows the construction of partitions of unity subordinate to such covers.^[22]^[23] Finally, \mathbb{R}^n satisfies the Lindelöf property: every open cover has a countable subcover. This follows directly from second countability, as any open cover can be refined by a countable basis subcollection that still covers the space, using the axiom of countable choice to select basis elements intersecting each cover set.^[24]

Key Structural Features

Open and Closed Sets

In the Euclidean topology on \mathbb{R}^n, a subset is open if it can be expressed as an arbitrary union of open balls, where an open ball of radius \epsilon > 0 centered at x \in \mathbb{R}^n consists of all points y satisfying \|y - x\| < \epsilon.^[25] This characterization arises from the metric-induced topology, ensuring that every point in an open set has a neighborhood entirely contained within it.^[26] Representative examples of open sets include open intervals (a, b) in \mathbb{R}, open disks \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \} in \mathbb{R}^2, and open half-spaces such as \{ x \in \mathbb{R}^n \mid x_1 > 0 \} in higher dimensions.^[25] These sets illustrate how openness captures regions without "boundary" points, allowing small perturbations around any interior point to remain inside the set. A subset of \mathbb{R}^n is closed if its complement is open.^[27] Equivalently, a closed set contains all its limit points, where a point p is a limit point of a set A if every open ball around p contains at least one point of A distinct from p.^[28] Examples of closed sets include closed balls \{ y \in \mathbb{R}^n \mid \|y - x\| \leq \epsilon \}, singletons \{x\}, the entire space \mathbb{R}^n, and finite unions of closed sets such as closed intervals [a, b] in \mathbb{R}.^[25] Many subsets are neither open nor closed; for instance, the rational numbers \mathbb{Q} as a subset of \mathbb{R} is dense (every open interval contains rationals and irrationals), so it has empty interior (no open ball lies entirely in \mathbb{Q}) and closure \mathbb{R} (it contains no isolated points).^[28] Similarly, the half-open interval [0, 1) in \mathbb{R} is not open (0 lacks a ball contained within it) and not closed (1 is a limit point not in the set).^[29] Another example is the unit circle \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \}, which is closed but not open, as points on it have no disk entirely on the circle.^[25] Clopen sets, which are both open and closed, exist trivially as the empty set and \mathbb{R}^n itself; due to the connectedness of Euclidean space, no nontrivial clopen subsets exist.^[30] The interior of a set A \subseteq \mathbb{R}^n, denoted \operatorname{int}(A), is the largest open set contained in A, consisting of all interior points p \in A such that some open ball around p lies entirely in A; this provides Euclidean intuition as points where distance to the complement is positive.^[28] The boundary of A, denoted \partial A, comprises points p \in \mathbb{R}^n that are limit points of both A and its complement, meaning every open ball around p intersects both A and \mathbb{R}^n \setminus A; intuitively, these are points at zero distance from both the set and its exterior.^[29] For example, the boundary of the open unit disk in \mathbb{R}^2 is the unit circle.^[25]

Continuity and Homeomorphisms

In the Euclidean topology on \mathbb{R}^n, equipped with the standard metric d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}, a function f: \mathbb{R}^m \to \mathbb{R}^n is defined to be continuous at a point x \in \mathbb{R}^m if for every \epsilon > 0, there exists \delta > 0 such that d(x, y) < \delta implies d(f(x), f(y)) < \epsilon for all y \in \mathbb{R}^m.^[31] This \epsilon-\delta condition captures the intuitive notion that small changes in the input produce small changes in the output, and it extends the classical definition from real analysis to higher-dimensional Euclidean spaces.^[31] The full function f is continuous if it satisfies this property at every point in its domain.^[31] Euclidean spaces are first-countable, meaning every point has a countable local basis of neighborhoods, which implies that the \epsilon-\delta definition of continuity is equivalent to sequential continuity.^[31] Specifically, f is continuous at x if and only if whenever a sequence (x_k) in \mathbb{R}^m converges to x, the sequence (f(x_k)) converges to f(x) in \mathbb{R}^n.^[31] This equivalence facilitates proofs in Euclidean topology, as sequences often provide a concrete way to verify limits and continuity without directly manipulating \epsilon and \delta.^[31] A homeomorphism between two Euclidean spaces is a bijective continuous function whose inverse is also continuous, preserving the topological structure induced by the metric.^[32] Examples include linear isometries such as translations, rotations, and reflections, which maintain distances and thus the open sets in the topology; more generally, any isometry of \mathbb{R}^n—a distance-preserving bijection—is a homeomorphism.^[32] However, not all dimensions are topologically equivalent: \mathbb{R}^n is not homeomorphic to \mathbb{R}^m for n \neq m, a consequence of Brouwer's invariance of domain theorem, which states that if U \subseteq \mathbb{R}^n is open and f: U \to \mathbb{R}^n is a continuous injection, then f(U) is open in \mathbb{R}^n.^[33] On compact subsets of Euclidean space, continuous functions exhibit stronger uniformity: a function f: K \to \mathbb{R}^n, where K \subseteq \mathbb{R}^m is compact, is uniformly continuous, meaning for every \epsilon > 0, there exists \delta > 0 such that d(x, y) < \delta implies d(f(x), f(y)) < \epsilon for all x, y \in K, independent of position.^[34] This follows from the Heine-Borel theorem, which characterizes compact subsets of \mathbb{R}^m as precisely the closed and bounded sets, allowing the \epsilon-\delta choices to be made globally via finite covers.^[34]

Special Properties and Theorems

Compactness

In Euclidean topology, compactness is a fundamental property that captures the idea of a space being "finite" in a topological sense, ensuring that every open cover has a finite subcover. This notion is particularly well-behaved in Euclidean spaces \mathbb{R}^n, where it aligns closely with metric properties like boundedness and closedness.^[35] The Heine-Borel theorem provides the precise characterization: a subset K \subseteq \mathbb{R}^n is compact if and only if it is closed and bounded.^[35] To outline the proof, first assume K is compact; then it is closed because its complement is open (complements of compact sets are open in \mathbb{R}^n), and bounded because otherwise an open cover by balls of increasing radius would lack a finite subcover.^[35] Conversely, if K is closed and bounded, scale and translate to assume K \subseteq [-1,1]^n; any open cover admits a Lebesgue number \delta > 0, and the boundedness allows covering by finitely many balls of radius \delta/2, each intersecting finitely many cover elements, yielding a finite subcover whose closedness in K prevents points from escaping.^[35] This equivalence fails in infinite-dimensional spaces but holds distinctly for finite-dimensional Euclidean topology.^[35] Euclidean spaces \mathbb{R}^n are locally compact, meaning every point has a compact neighborhood, such as a closed ball \overline{B(x, r)} for small r > 0, which is compact by the Heine-Borel theorem.^[36] Examples of non-compact subsets include the open unit ball \{x \in \mathbb{R}^n : \|x\| < 1\}, which is bounded but not closed, and unbounded sets like \mathbb{R}^n itself, or dense non-closed sets like \mathbb{Q}^n.^[35] A key consequence is the Bolzano-Weierstrass theorem: every bounded sequence in \mathbb{R}^n has a convergent subsequence, with the limit in the closure due to completeness.^[37] This follows from compactness of the closed ball containing the sequence's range.^[37] Applications include the extreme value theorem: a continuous function f: K \to \mathbb{R} on a compact K \subseteq \mathbb{R}^n attains its maximum and minimum, as the image f(K) is compact (continuous images preserve compactness) and thus closed and bounded in \mathbb{R}.^[38] This ensures extrema exist without boundary checks beyond the set itself.^[38]

Connectedness

In Euclidean topology, the space \mathbb{R}^n equipped with the standard topology is connected, meaning it cannot be expressed as the union of two nonempty disjoint open sets.^[39] This property holds for all n \geq 1, as any attempt to separate \mathbb{R}^n into such sets would contradict the density and unboundedness of the rational points or the behavior of continuous functions on intervals.^[40] Furthermore, connectedness in \mathbb{R}^n is equivalent to path-connectedness, where any two points can be joined by a continuous path; this equivalence arises because connected open subsets of \mathbb{R}^n are path-connected, with paths constructed as polygonal lines consisting of straight line segments between points, avoiding obstacles in the complement.^[41]^[40] Open sets in \mathbb{R}^n are locally path-connected, possessing a basis of open balls that are themselves path-connected due to their convexity.^[41] In this context, the connected components of \mathbb{R}^n are the entire space itself, as it forms a single connected piece; however, in discrete subspaces induced by the subspace topology, such as finite sets, the connected components reduce to singletons, reflecting the isolated nature of points without limiting neighborhoods.^[41] For example, in \mathbb{R}, any interval (open, closed, or half-open) is connected, while a union of two disjoint nonempty intervals is disconnected, as each interval serves as a clopen set in the subspace topology.^[42] In higher dimensions, the sphere S^{n-1} = \{ x \in \mathbb{R}^n : \|x\| = 1 \} is connected for n \geq 2, as it cannot be separated by open sets in the subspace topology, though S^0 consists of two disconnected points.^[43] A key consequence of connectedness is the intermediate value theorem for continuous functions: if f: K \to \mathbb{R} is continuous on a connected subset K \subseteq \mathbb{R}^n and f attains values a < b at points in K, then f attains every value between a and b.^[40] This follows from the preservation of connectedness under continuous maps, ensuring the image f(K) is connected and thus an interval in \mathbb{R}. Additionally, \mathbb{R}^n exhibits arcwise connectedness, where points in convex subsets—such as open balls or the entire space—are joined by straight-line arcs, leveraging the linearity of the Euclidean metric.^[44]

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[PDF] Spaces that are connected but not path connected - Keith Conrad
The most fundamental example of a connected set is the interval [0,1], or more generally any closed or open interval in R. Most reasonable-looking spaces that ...
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[PDF] Point-Set Topology II
Sep 14, 2010 · By the Intermediate Value Theorem, f ◦ γ has a zero, and so f has a zero. It follows that Sn is connected and R r {0} is not path-connected.
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[PDF] Notes on Connectivity Introduction 1 Arcwise connectedness 2 ...
The set C(x) is called the connected component, or just the component, of x. ... rational numbers, in the relative topology of R, furnishes a striking example of ...