Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions
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Show that lower semicontinuous function $f:Xrightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions.
My attempt: I don't know how to approximate $f(x)$ to within $[f(x)-1/n,f(x)]$ by $h_n(x)$ which is a linear combination of characteristic function of open sets, using lower semicontinuity of $f$.
Can anyone help?
general-topology analysis semicontinuous-functions
edited Sep 1 at 0:27
Martin Sleziak
43.7k6113260
asked May 13 '15 at 4:02
scyphi
253
add a comment |Â
up vote
4
down vote
favorite
Show that lower semicontinuous function $f:Xrightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions.
My attempt: I don't know how to approximate $f(x)$ to within $[f(x)-1/n,f(x)]$ by $h_n(x)$ which is a linear combination of characteristic function of open sets, using lower semicontinuity of $f$.
Can anyone help?
general-topology analysis semicontinuous-functions
edited Sep 1 at 0:27
Martin Sleziak
43.7k6113260
asked May 13 '15 at 4:02
scyphi
253
1
Now sure how to proceed with your idea. But from a post a couple years back... Let $f_k(x)=inff(y)+kd(x,y):yin X.$
– matt biesecker
May 13 '15 at 5:00
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Show that lower semicontinuous function $f:Xrightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions.
My attempt: I don't know how to approximate $f(x)$ to within $[f(x)-1/n,f(x)]$ by $h_n(x)$ which is a linear combination of characteristic function of open sets, using lower semicontinuity of $f$.
Can anyone help?
general-topology analysis semicontinuous-functions
edited Sep 1 at 0:27
Martin Sleziak
43.7k6113260
asked May 13 '15 at 4:02
scyphi
253
Show that lower semicontinuous function $f:Xrightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions.
My attempt: I don't know how to approximate $f(x)$ to within $[f(x)-1/n,f(x)]$ by $h_n(x)$ which is a linear combination of characteristic function of open sets, using lower semicontinuity of $f$.
Can anyone help?
general-topology analysis semicontinuous-functions
general-topology analysis semicontinuous-functions
edited Sep 1 at 0:27
Martin Sleziak
43.7k6113260
asked May 13 '15 at 4:02
scyphi
253
edited Sep 1 at 0:27
Martin Sleziak
43.7k6113260
asked May 13 '15 at 4:02
scyphi
253
edited Sep 1 at 0:27
Martin Sleziak
43.7k6113260
edited Sep 1 at 0:27
Martin Sleziak
43.7k6113260
edited Sep 1 at 0:27
Martin Sleziak
43.7k6113260
43.7k6113260
asked May 13 '15 at 4:02
scyphi
253
asked May 13 '15 at 4:02
scyphi
253
asked May 13 '15 at 4:02
scyphi
253
253
1
Now sure how to proceed with your idea. But from a post a couple years back... Let $f_k(x)=inff(y)+kd(x,y):yin X.$
– matt biesecker
May 13 '15 at 5:00
add a comment |Â
1
Now sure how to proceed with your idea. But from a post a couple years back... Let $f_k(x)=inff(y)+kd(x,y):yin X.$
– matt biesecker
May 13 '15 at 5:00
1
1
Now sure how to proceed with your idea. But from a post a couple years back... Let $f_k(x)=inff(y)+kd(x,y):yin X.$
– matt biesecker
May 13 '15 at 5:00
Now sure how to proceed with your idea. But from a post a couple years back... Let $f_k(x)=inff(y)+kd(x,y):yin X.$
– matt biesecker
May 13 '15 at 5:00
add a comment |Â
1 Answer
1
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oldest
votes
up vote
2
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accepted
This is a quotation from "General Topology" by Ryszard Engelking:
answered May 16 '15 at 8:32
Alex Ravsky
36.6k32075
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
This is a quotation from "General Topology" by Ryszard Engelking:
answered May 16 '15 at 8:32
Alex Ravsky
36.6k32075
add a comment |Â
up vote
2
down vote
accepted
This is a quotation from "General Topology" by Ryszard Engelking:
answered May 16 '15 at 8:32
Alex Ravsky
36.6k32075
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
This is a quotation from "General Topology" by Ryszard Engelking:
answered May 16 '15 at 8:32
Alex Ravsky
36.6k32075
This is a quotation from "General Topology" by Ryszard Engelking:
answered May 16 '15 at 8:32
Alex Ravsky
36.6k32075
answered May 16 '15 at 8:32
Alex Ravsky
36.6k32075
answered May 16 '15 at 8:32
Alex Ravsky
36.6k32075
answered May 16 '15 at 8:32
Alex Ravsky
36.6k32075
36.6k32075
add a comment |Â
add a comment |Â
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1
Now sure how to proceed with your idea. But from a post a couple years back... Let $f_k(x)=inff(y)+kd(x,y):yin X.$
– matt biesecker
May 13 '15 at 5:00