# Compactness of topological space

• May 23rd 2012, 07:07 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, 08:43 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 $O_{i_0}$).
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9806976914405823, "perplexity": 142.87323675046568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802772751.143/warc/CC-MAIN-20141217075252-00164-ip-10-231-17-201.ec2.internal.warc.gz"}