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
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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