Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right

Let $(X,\mathcal{T})$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

• An open cover of $(X,\mathcal{T})$ is a family $\mathcal{U}\subseteq \mathcal{T}$ of open sets with $X=\bigcup \mathcal{U}$, where $\bigcup \mathcal{U}=\{x\in X:x\in U\text{ for some }U\in \mathcal{U}\}$.
• A subcover of $\mathcal{U}$ is a subfamily $\mathcal{V}\subseteq \mathcal{U}$ that is itself an open cover.
• A family $\mathcal{V}$ of sets is finite when $\mathcal{V}=\mathrm{\varnothing }$ or there are $n\in \mathbb{N}$ and sets $V_0,\dots ,V_n$ with $\mathcal{V}=\{V_0,\dots ,V_n\}$; repetitions in the list are allowed and harmless.
• $(X,\mathcal{T})$ is compact when every open cover of it has a finite subcover: for every open cover $\mathcal{U}$, either $X=\mathrm{\varnothing }$ and the empty subfamily covers it, or there are $n\in \mathbb{N}$ and $U_0,\dots ,U_n\in \mathcal{U}$ with $X=U_0\cup \cdots \cup U_n\mathrm{.}$
• A subset $A\subseteq X$ is a compact subset of $X$ when the subspace $(A,{\mathcal{T}}_A)$ is a compact topological space, ${\mathcal{T}}_A$ being the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

Compactness of a subset is defined intrinsically, and only intrinsically. The last clause speaks about the subspace $(A,{\mathcal{T}}_A)$ and its own open sets, not about families of open subsets of the ambient $X$. The two readings do agree, but that is a theorem and not a convention: it is A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, and no item of this library may use the ambient reading without citing it. Taking the intrinsic reading makes compactness a property of the topological space $(A,{\mathcal{T}}_A)$ alone. Hence it is preserved when $A$ is embedded homeomorphically as a subspace, or when another ambient space induces the same topology on $A$; it need not be preserved if the induced topology changes. This is exactly the convention already fixed for metric spaces by Open cover, subcover, compact metric space, and compact subset of a metric space, and the agreement of that definition with this one is For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide.

The empty space is compact, since the empty subfamily of any family covers it; this is the reason the clause above is written with the two cases. Every space listed as $\{x_0,\dots ,x_n\}$ is compact too: given a cover, each $x_i$ lies in some member, and finitely many members named in this way already cover. So every finite space is compact, whatever its topology, and in particular the discrete topology on a finite set is compact while the discrete topology on an infinite set is not (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

The finiteness convention. "Finite" above is the listing form. It agrees with the definition of finiteness by equinumerosity with a natural number (Finite, countably infinite, countable, uncountable), in both directions, and the agreement is the one already discharged in Open cover, subcover, compact metric space, and compact subset of a metric space: a nonempty set equinumerous with $m\ge 1$ is listable, and a set listed as $\{a_0,\dots ,a_n\}$ injects into $\sigma (n)$ by sending $x$ to the least $i\le n$ with $a_i=x$. Neither direction uses a choice principle; the second selects nothing, taking a least index instead.

Quasicompact is not used here. Some authors, following Bourbaki, reserve compact for a space that is both compact in the above sense and Hausdorff, and call the open-cover condition alone quasicompact. This library follows the more widely adopted convention: compact means the open-cover condition and nothing more, and a Hausdorff hypothesis is always written out. The fork is recorded in The quasicompact convention, why compactness of a subset is read intrinsically here, and what each result on this page costs in choice.

Why open covers rather than covers by arbitrary sets. Nothing in the definition would break if $\mathcal{U}$ were allowed to consist of arbitrary subsets of $X$, but the resulting notion would be uninteresting: every space is covered by its singletons, and only a finite space would survive. Openness of the members is what makes the condition a genuine restriction, and it is what A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it has to keep track of when the ambient space changes.

A warning about the word "cover". A family may cover $A\subseteq X$ without being a family of subsets of $A$: the members are open subsets of $X$ and their union merely contains $A$. That is the ambient reading, and it is a different statement from "$\mathcal{U}$ is an open cover of the space $(A,{\mathcal{T}}_A)$", whose members are open subsets of $A$. Which of the two is meant is written out everywhere on this page.