TITLE: What is the neighborhood-definition of a compact space?
QUESTION [2 upvotes]: Looking through some articles on topology, I see a neighborhood definition of 'topological space' and a neighborhood definition of 'Hausdorff space', but do not see a neighborhood definition of 'compact space'.
The 'open cover' definition of compact space states it is where each open cover has a finite subcover. Is this equivalent to a space being compact when every union of neighborhoods has a finite subcover consisting of neighborhoods?
What is the neighborhood definition of a compact space?

REPLY [0 votes]: Let $T$ be a topological space.

Neighborhood definition of open set:

A set $S \subset T$ is open means every $s \in S$ has a neighborhood $N_s$ contained in $S$.
That is, $\forall s \in S$ $\exists N_s \subset S$.

Open set definition of open cover:

Given a set $S$ in a topological space, an open cover of $S$ is a family of open sets $O_i |_{i \in I}$ whose union contains $S$. That is; $O_i |_{i\in I}$ is an open cover of $S$ means $O_i$ is open $\forall i \in I$ and $S \subset \bigcup_{i \in I} O_i$.

Open cover definition of compact space:

$T$ is compact means every open cover has a finite subcover. That is, given an open cover $O_i |_{i\in I} \subset T$ $\exists$ finite $J \subset I$ such that $O_j |_{j\in J} \subset T$.

Thus, neighborhoods form a definition of open set, which forms the definition of open cover, which forms the definition of compact space.
To describe a compact space directly in terms of neighborhoods, we might say:
A topological space $T$ is compact means given a covering family $T \subset O|_{i \in I}$ where $O_i$ contains a neighborhood for each of its points, there is a finite subfamily that also covers $T$. That is, Given $O|_{i \in I}$ where $T \subset O|_{i \in I}$ and $\forall x \in O_i$ $\exists N_x \in O_i$, $\exists$ finite $J \subset I$ such that $T \subset O|_{j \in J}$.
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