compactness and accumulation points of nets

Theorem.

A topological space $X$ is compact if and only if every net in $X$ has an accumulation point.

Proof.

Suppose $X$ is compact and let ${(x_{\alpha })}_{\alpha \in A}$ be a net in $X$. For each $\alpha \in A$, put $E_{\alpha }=\{x_{\beta }:\beta \ge \alpha \}$; the collection $\{\overline{E_{\alpha }}:\alpha \in A\}$ of closed subsets of $X$ has the finite intersection property, for given ${\alpha }_1,\mathrm{\dots },{\alpha }_n\in A$, because $A$ is directed, there exists $\beta \in A$ satisfying $\beta \ge {\alpha }_i$ for each $i\in \{1,\mathrm{\dots },n\}$, so that $x_{\beta }\in {\bigcap }_{i=1}^n\overline{E_{{\alpha }_i}}$. Therefore, by compactness, ${\bigcap }_{\alpha \in A}\overline{E_{\alpha }}\ne \mathrm{\varnothing }$; let $x$ be a point of this intersection. If $U$ is any open subset of $X$ and $\alpha \in A$, then because $x\in \overline{E_{\alpha }}$, $E_{\alpha }\cap U\ne \mathrm{\varnothing }$, and thus there exists $\beta \ge \alpha \in A$ for which $x_{\beta }\in U$. It follows that $x$ is an accumulation point of $(x_{\alpha })$. For the converse, assume that $X$ fails to be compact, and let $\{U_i:i\in I\}$ be an open cover of $X$ with no finite subcover. If $B$ is the set of finite subsets of $I$, then $B$ is directed by inclusion. For each set $S\in B$, let $x_S$ be a point in the complement of ${\bigcup }_{i\in S}{U}_i$. We contend that the net ${(x_S)}_{S\in B}$ has no accumulation points; indeed, given $x\in X$, we may select $i_0\in I$ such that $x\in U_{i_0}$; if $S\in B$ is such that $i_0\in S$, that is, if $S\ge \{i_0\}$, then by construction, $x_S\notin U_{i_0}$, establishing our contention.∎

Corollary.

The following conditions on a topological space $X$ are equivalent:

1. $X$ is compact;

2. every net in $X$ has an accumulation point;

3. every net in $X$ has a convergent subnet;

Proof.

The preceding theorem establishes the equivalence of (1) and (2), while that of (2) and (3) is established in the entry on accumulation points and convergent subnets.∎