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If S is a compact subset of R and T is a closed subset of S,then T is compact. (1) Prove this using the definition of compactness.

Can somebody prove it? I think we should select a open cover of S randomly, and then we should think about the set S-T. Is S-T open in R? I don't know how to continue?

MJD
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python3
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  • It doesn't matter if $S-T$ is open in $R$. The definition of compactness is internal: A topological space $X$ is compact if any cover of $X$ by sets which are open in $X$ contains an open subcover of $X$. – Avi Steiner Mar 18 '14 at 01:05
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    what can you say about $\lbrace T^c \rbrace \cup \lbrace U_n ;,n\in {\rm I!N,}\rbrace$ where $(U_i)_{\scriptsize {i \in I}}$ a family of open (open for the topology of S) whose union contains S? –  Mar 18 '14 at 01:05
  • Next time please choose a more descriptive title for your question. – MJD Mar 18 '14 at 01:08
  • I know S-T is open in S, but Why R-T is open ? – python3 Mar 18 '14 at 01:15
  • What is the definition of a closed set in $\Bbb R$, can you tell me?? – Ishfaaq Mar 18 '14 at 01:17
  • T is closed if and only if R-T is open. – python3 Mar 18 '14 at 01:19
  • @tiandiao123: Exactly! – Ishfaaq Mar 18 '14 at 01:20

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Consider any open cover $G_{\lambda}$ of $T$. Then if $S \subseteq G_{\lambda}$ too there is a finite covering of $S$ using sets from $G_{\lambda}$ which also contains $T$ and hence is a finite covering of $T$. Suppose $S \not \subseteq G_{\lambda}$. Then consider $ G_{\lambda} \cup T^C $ which is an open covering of $S$ since $T$ is closed and $T^C$ is an open set. Then again since $S$ is compact we have that there is a finite covering of $S$ using sets in $G_{\lambda} \cup T^C $. Removing $T^C$ if it was part of this finite covering we have a finite covering of $T$. Hence $T$ is compact.

Ishfaaq
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    The question was a good example of a question for which it easy to give useful hints, and probably considerably better and/or more useful than writing down the complete solution –  Mar 18 '14 at 01:09
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    @Julien: True. This is the first time I came across the problem too. Guess I got excited after solving it. – Ishfaaq Mar 18 '14 at 01:11
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    For those of us reading later having the whole answer spelled out is much more useful than parsing a bunch of hints. This is a Q and A site not a homework coaching site. – Bob Woodley Jul 14 '15 at 19:37
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    In Munkres's book I read something similar to " removing $T^c$ if it was part of this finite covering". However, if not removing it, it is still a finite covering of T. – whitegreen Sep 04 '20 at 00:18