Infinite Particular Point Space is not Compact

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \tau_p}\right)$ be an infinite particular point space.


Then $T$ is not compact.


Proof

Consider the open cover of $T$:

$\mathcal C = \left\{{\left\{{x, p}\right\}: x \in S, x \ne p}\right\}$

As $S$ is infinite, then so is $\mathcal C$, as we can set up a bijection from $\phi: S \setminus \left\{{p}\right\} \leftrightarrow \mathcal C$:

$\forall x \in S \setminus \left\{{p}\right\}: \phi \left({x}\right) = \left\{{x, p}\right\}$

Hence $\mathcal C$ has no finite subcover.

The result follows by definition of compactness.

$\blacksquare$


Sources