I Confused by proof in Hocking and Young

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yossell
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Am trying to learn topology (again) and failing.

Lemma 2.8, p43 of Hocking and Young states: Let ##a## and ##b## be distinct points of a compact Hausdorff space ##S## and let ##\{H_\alpha \}## be a collection of closed sets simply ordered by inclusion. If each ##H_\alpha## is contains both ##a## and ##b## but is not the union of two separated sets,one containing ##a## and the other containing ##b##,then the intersection ##\cap_\alpha H_\alpha## also has this property.

Proof: Let ##H = \cap_\alpha H_\alpha## and suppose that ##H## is the union of two separated sets ##A## and ##B##, with ##a## in ##A## and ##b## in ##B##. Since H is closed and ##A## and ##B## are closed in ##H##, it follows that ##A## and ##B## are closed in the space ##S## and...'

It's that last 'it follows' that is losing me -- I can't see why it follows that they're closed in ##S##. I'm missing something trivial, but I don't know what. Any suggestions would be appreciated.

MENTOR note: replaced all $ with double # to render properly under mathjax
 
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This is trivial. There are two closed sets ## A ## and ## B ## and they will always be closed, no matter where they are placed, in ## H ## or in ## S ##. The only important thing is that the sets ## A ## and ## B ## are closed because of the Axiom C1.
 
Why are ##A,B## closed in ##H##?

If so, then they are closed in ##S,## too:

The boundary of ##A## in ##S## is contained in ##H## and the boundary of ##H## in ##S##. But ##H\subseteq S## is closed, so its boundary in ##S## is fully contained in ##H##. Therefore, the boundary of ##A## in ##S## is fully contained in ##H## and since ##A\subseteq H## is closed, fully contained in ##A.## This means that ##A## is closed in ##S.## For short: you cannot gain additional boundary points of ##A## from searching in ##S## since all those points are already in ##H## and its boundary in ##S,## and thus in ##A.##
 
I took a quick look at that book and wondered why you were interested in such a tedious approach to topology. Then I noticed some very nice theorems on pages 54-55 with simple characterizations of a circle and an interval. E.g. apparently any compact connected metric space which is disconnected by the removal of any point, except for two of them, is homeomorphic to a closed interval. And such a c.c. metric space that is disconnected by the removal of any two points, is apparently homeomorphic to a circle. nice.
 
By " Simply ordered by inclusion", you mean a total order , i.e., for ## O, O'## closed, either ## O \subset O' ## or ##O' \subset O##?
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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