# Thread: Compactness of topological space

1. ## Compactness of topological space

Hey.

My assignment says the following:

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

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

(2) SHow that $\displaystyle (X^* , \tau^* )$ 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

2. ## Re: Compactness of topological space

Take an open over of $\displaystyle X^*$, say $\displaystyle (O_i)_{i\in I}$. For some $\displaystyle i_0$, we have $\displaystyle \alpha\in O_{i_0}$, hence necessarily $\displaystyle O_{i_0}$ is of the form $\displaystyle X^*\setminus U_{i_0}$ where $\displaystyle U_{i_0}$ is a closed compact subset of $\displaystyle X$. $\displaystyle U_{i_0}$ is covered by $\displaystyle (O_i)_{i\in I}$ hence you can extract a finite subcover and you are done (just add $\displaystyle O_{i_0}$).