Upper semicontinuous function as a poinwise limit of continuous fuctions
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The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous functions. I was able to find the proof (Bourbaki, General Topology, part II) only for the case when X is perfectly normal. Is the general statement above true, and if it is where can I find a proof?
general-topology pointwise-convergence semicontinuous-functions
edited Dec 24 '18 at 22:15
Alex Ravsky
43.6k32584
asked Dec 24 '18 at 19:55
Arkady KitoverArkady Kitover
362
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add a comment |
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The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous functions. I was able to find the proof (Bourbaki, General Topology, part II) only for the case when X is perfectly normal. Is the general statement above true, and if it is where can I find a proof?
general-topology pointwise-convergence semicontinuous-functions
edited Dec 24 '18 at 22:15
Alex Ravsky
43.6k32584
asked Dec 24 '18 at 19:55
Arkady KitoverArkady Kitover
362
$endgroup$
add a comment |
$begingroup$
The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous functions. I was able to find the proof (Bourbaki, General Topology, part II) only for the case when X is perfectly normal. Is the general statement above true, and if it is where can I find a proof?
general-topology pointwise-convergence semicontinuous-functions
edited Dec 24 '18 at 22:15
Alex Ravsky
43.6k32584
asked Dec 24 '18 at 19:55
Arkady KitoverArkady Kitover
362
$endgroup$
The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous functions. I was able to find the proof (Bourbaki, General Topology, part II) only for the case when X is perfectly normal. Is the general statement above true, and if it is where can I find a proof?
general-topology pointwise-convergence semicontinuous-functions
general-topology pointwise-convergence semicontinuous-functions
edited Dec 24 '18 at 22:15
Alex Ravsky
43.6k32584
asked Dec 24 '18 at 19:55
Arkady KitoverArkady Kitover
362
edited Dec 24 '18 at 22:15
Alex Ravsky
43.6k32584
asked Dec 24 '18 at 19:55
Arkady KitoverArkady Kitover
362
edited Dec 24 '18 at 22:15
Alex Ravsky
43.6k32584
edited Dec 24 '18 at 22:15
Alex Ravsky
43.6k32584
edited Dec 24 '18 at 22:15
Alex Ravsky
43.6k32584
43.6k32584
asked Dec 24 '18 at 19:55
Arkady KitoverArkady Kitover
362
asked Dec 24 '18 at 19:55
Arkady KitoverArkady Kitover
362
asked Dec 24 '18 at 19:55
Arkady KitoverArkady Kitover
362
362
add a comment |
add a comment |
1 Answer
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See below Problem 1.7.15.c from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989).
answered Dec 24 '18 at 22:10
Alex RavskyAlex Ravsky
43.6k32584
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
See below Problem 1.7.15.c from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989).
answered Dec 24 '18 at 22:10
Alex RavskyAlex Ravsky
43.6k32584
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add a comment |
$begingroup$
See below Problem 1.7.15.c from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989).
answered Dec 24 '18 at 22:10
Alex RavskyAlex Ravsky
43.6k32584
$endgroup$
add a comment |
$begingroup$
See below Problem 1.7.15.c from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989).
answered Dec 24 '18 at 22:10
Alex RavskyAlex Ravsky
43.6k32584
$endgroup$
See below Problem 1.7.15.c from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989).
answered Dec 24 '18 at 22:10
Alex RavskyAlex Ravsky
43.6k32584
answered Dec 24 '18 at 22:10
Alex RavskyAlex Ravsky
43.6k32584
answered Dec 24 '18 at 22:10
Alex RavskyAlex Ravsky
43.6k32584
answered Dec 24 '18 at 22:10
Alex RavskyAlex Ravsky
43.6k32584
43.6k32584
add a comment |
add a comment |
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