# 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 $E_{\alpha}=\{x_{\beta}:\beta\geq\alpha\}$; the collection $\{\overline{E_{\alpha}}:\alpha\in A\}$ of closed subsets of $X$ has the finite intersection property, for given $\alpha_{1},\ldots,\alpha_{n}\in A$, because $A$ is directed, there exists $\beta\in A$ satisfying $\beta\geq\alpha_{i}$ for each $i\in\{1,\ldots,n\}$, so that $x_{\beta}\in\bigcap_{i=1}^{n}\overline{E_{\alpha_{i}}}$. Therefore, by compactness, $\bigcap_{\alpha\in A}\overline{E_{\alpha}}\neq\emptyset$; let $x$ be a point of this intersection. If $U$ is any open subset of $X$ and $\alpha\in A$, then because $x\in\overline{E_{\alpha}}$, $E_{\alpha}\cap U\neq\emptyset$, and thus there exists $\beta\geq\alpha\in A$ 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 $S\geq\{i_{0}\}$, then by construction, $x_{S}\notin U_{i_{0}}$, establishing our contention. ∎

###### Corollary.

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

1. 1.

$X$ is compact;

2. 2.

every net in $X$ has an accumulation point;

3. 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 CompactnessAndAccumulationPointsOfNets 2013-03-22 18:37:50 2013-03-22 18:37:50 azdbacks4234 (14155) azdbacks4234 (14155) 7 azdbacks4234 (14155) Theorem msc 54A20 Net Compact AccumulationPointsAndConvergentSubnets