Compact space - Intuition and Basic Examples

Understanding Compactness Introduction: What is Compactness? Compactness is one of the most important concepts in topology and analysis. Intuitively, a space is compact if it is "small" and "closed up" in a specific mathematical sense. More precisely, compactness generalizes the familiar idea of a closed and bounded subset of Euclidean space. The core intuition behind compactness is this: every infinite sequence of points in a compact space has a convergent subsequence that stays within the space. This means there are no "escape routes"—points cannot drift off to infinity or approach a boundary that lies outside the space. Why Compactness Matters Compactness is useful because it guarantees certain desirable properties. For example: Continuous functions on compact spaces achieve their maximum and minimum values The image of a compact space under a continuous function is itself compact Many existence theorems in analysis rely on compactness Simple Illustrations Let's ground this intuition with concrete examples from the real line: The real line ℝ is not compact. Consider the sequence 1, 2, 3, 4, ... This is an infinite sequence of points in ℝ, but no subsequence of it converges to any real number—the sequence drifts toward infinity. This shows ℝ fails to be compact. A closed interval like [0, 1] is compact. Any infinite selection of points from this interval has a subsequence that converges to a point within the interval. You cannot pick points that escape to infinity or jump outside the interval. The extended real line ℝ ∪ {−∞, +∞} (which includes both infinities) is compact. By including the infinities as actual points, we ensure that sequences drifting toward infinity still converge to a point within the space. The image above illustrates three different subsets of the real line. Interval B (the closed interval) is compact, while intervals A and C extend outward unboundedly and are therefore not compact. Basic Examples of Compact and Non-Compact Spaces Finite Spaces Every finite topological space is compact. This is straightforward: if a space has only finitely many points, then any open cover contains at least one open set for each point, so we can always extract a finite subcover by picking at least one set for each point. Intervals and Disks Closed intervals like [a, b] are compact. This is one of the fundamental examples you should know. Closed disks (including their boundaries) are compact. More generally, any closed and bounded subset of Euclidean space is compact. Open disks and open intervals are not compact. For example, the open interval (0, 1) is not compact because we can construct sequences that approach the endpoints 0 or 1 without being able to converge to a point inside (0, 1). The endpoints lie outside the space. Spheres and Higher-Dimensional Balls Spheres are compact. A sphere (the surface of a ball, not including its interior) is compact in Euclidean space. A sphere with one point removed is not compact. For instance, if you remove the north pole from a 2-sphere, sequences of points can approach the missing north pole without converging to any point remaining in the space. The closed unit ball $B = \{x \in \mathbb{R}^n : \|x\| \leq 1\}$ is compact. The closed ball is closed and bounded, so it is compact. The open unit ball $B = \{x \in \mathbb{R}^n : \|x\| < 1\}$ is not compact. Points can approach the boundary without the space including the boundary. Unbounded Sets Are Not Compact Lines, planes, and half-infinite intervals like [0, ∞) are not compact. The defining feature is that points can "escape to infinity." You can construct sequences that drift toward infinity without any convergent subsequence. For example, in [0, ∞), the sequence 1, 2, 3, 4, ... has no convergent subsequence. The Heine–Borel Theorem One of the most important classical results is the Heine–Borel theorem: A subset of $\mathbb{R}^n$ (with the standard topology) is compact if and only if it is closed and bounded. This theorem is crucial: it gives you a simple practical criterion for checking compactness in Euclidean spaces. A set must be both closed (contain its boundary) and bounded (not extend to infinity) to be compact. Examples: $[0, 1]$ is closed and bounded, so it is compact ✓ $(0, 1)$ is bounded but not closed, so it is not compact ✗ $[0, \infty)$ is closed but not bounded, so it is not compact ✗ The set of rational numbers in $[0, 1]$ is bounded but not closed (it's missing its irrational endpoints), so it is not compact ✗ The entire real line ℝ is neither closed in a bounded set nor bounded, so it is not compact. Key Results and Properties Tychonoff's Theorem Tychonoff's theorem is a powerful result that extends compactness to infinite-dimensional spaces: The product of arbitrarily many compact spaces is compact. A classical example is the Hilbert cube, which is the product of countably many copies of the unit interval [0, 1]: $$[0,1] \times [0,1] \times [0,1] \times \cdots$$ By Tychonoff's theorem, this infinite-dimensional product space is compact, even though it cannot be embedded in any finite-dimensional Euclidean space. <extrainfo> Specialized Compact Spaces The Cantor set is compact. The Cantor set, constructed by repeatedly removing middle thirds from the unit interval, is a compact fractal set. In fact, every non-empty compact metric space is a continuous image of the Cantor set. Spaces with the cofinite topology are compact. A space has the cofinite topology if its open sets are the empty set and all sets whose complements are finite. Any space equipped with this topology is compact. The extended real line is compact. We can establish a homeomorphism between the extended real line $\mathbb{R} \cup \{-\infty, +\infty\}$ and the closed interval $[-1, 1]$, which shows it is compact. </extrainfo> Non-Compact Examples No infinite discrete space is compact. In a discrete space (where every subset is open), the collection of all singletons {x} forms an open cover. Since there are infinitely many singletons, no finite subcover can cover all points. The real line is not compact. One way to see this: the cover of open intervals {(n − 1, n + 1) : n ∈ ℤ} covers ℝ, but no finite subcollection covers the entire line. <extrainfo> Arzelà–Ascoli Theorem In the space of continuous real-valued functions on a compact Hausdorff space, the Arzelà–Ascoli theorem characterizes when a subset is relatively compact (meaning its closure is compact). A subset of continuous functions is relatively compact if and only if it is equicontinuous and pointwise bounded. This is an important result in functional analysis and the theory of function spaces, but it represents a more specialized application of compactness. </extrainfo>