Compactness and connectedness

An open cover of a space is a family of open sets whose union contains every point. The space is compact when every such cover has a finite subcover. The word every makes this a global property. For the interval (0,1), the sets $(1/n,1)$ for integers n≥2 form an open cover in its subspace topology. Any finite selection has a largest n and misses sufficiently small positive points. Boundedness therefore does not establish compactness.

In finite-dimensional Euclidean space with its usual topology, the Heine-Borel theorem says that a subset is compact exactly when it is closed and bounded. The ambient-space qualification is essential. In an infinite-dimensional Hilbert space, the closed unit ball is bounded and closed but not norm-compact: orthonormal unit vectors remain distance √2 apart and have no norm-convergent subsequence. In metric spaces compactness is equivalent to sequential compactness, and also to completeness plus total boundedness. Boundedness alone is weaker than total boundedness.

A separation writes a space as the union of two disjoint, nonempty subsets that are both open in that space. Connected spaces admit no such separation. The phrase in that space refers to the subspace topology: the intervals [0,1] and [2,3] are open relative to their union, despite not being open in the entire real line. Their union is disconnected. Intervals in the real line are connected because a hypothetical separation would contradict the order-completeness underlying intermediate-value arguments.

A path is a continuous map from [0,1] into a space with specified endpoints. Path-connectedness implies connectedness because an interval cannot cross a separation continuously. The converse fails for general topological spaces. It does hold for connected manifolds, which are locally path-connected. This distinction prevents a drawing from doing too much work: a complicated connected set need not supply a continuous route between every pair of its points, even when its points appear to crowd together.

A continuous image of a compact space is compact: pull an open cover backward, take a finite subcover in the domain, and push the resulting coverage forward. A continuous image of a connected space is connected: a separation of the image would pull back to a separation of the domain. These are different proofs with the same useful strategy of working with preimages. A continuous real-valued function on a nonempty compact space consequently attains both a maximum and a minimum.

For a compact interval, the extreme-value theorem and intermediate-value theorem combine to make the image a closed interval. Removing compactness may lose attained endpoints; removing connectedness may leave gaps. For example, the identity on (0,1) has a connected image but no maximum. A function on two isolated points has a compact image but need not assume intermediate values. Before invoking a theorem, identify which hypothesis performs the desired job instead of treating all pleasant domain properties as interchangeable.