# Compactness of topological space

• May 23rd 2012, 06:07 PM
m112358
Compactness of topological space
Hey.

My assignment says the following:

Let $X$ be a topolical space with $\tau$ as its topology. Let $\infty$ be a point not in $X$. Let $X^* = X \cup \{ \infty \}$.
Let $\tau^* = \tau \cup \{ U \in X | X^* \setminus U$ is a closed, compact subset of $X \}$.

(1) Prove that $\tau^*$ is a topology on $X^*$. (I have already done this)

(2) SHow that $(X^* , \tau^* )$ is compact.

I dont know how to show number 2? Could anyone give me a hint or some advice on how to approach this?

Thanks a million.

/Morten
• May 24th 2012, 07:43 AM
girdav
Re: Compactness of topological space
Take an open over of $X^*$, say $(O_i)_{i\in I}$. For some $i_0$, we have $\alpha\in O_{i_0}$, hence necessarily $O_{i_0}$ is of the form $X^*\setminus U_{i_0}$ where $U_{i_0}$ is a closed compact subset of $X$. $U_{i_0}$ is covered by $(O_i)_{i\in I}$ hence you can extract a finite subcover and you are done (just add $O_{i_0}$).