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. ∎
Title  compactness and accumulation points of nets 

Canonical name  CompactnessAndAccumulationPointsOfNets 
Date of creation  20130322 18:37:50 
Last modified on  20130322 18:37:50 
Owner  azdbacks4234 (14155) 
Last modified by  azdbacks4234 (14155) 
Numerical id  7 
Author  azdbacks4234 (14155) 
Entry type  Theorem 
Classification  msc 54A20 
Related topic  Net 
Related topic  Compact 
Related topic  AccumulationPointsAndConvergentSubnets 