mathlib4
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feat(Topology/UrysohnsLemma): add variations of Urysohn's lemma
add variations of Urysohn's lemma:
- In a normal space
X
, for a closed sett
and an open sets
such thatt ⊆ s
, there is a continuous functionf
supported ins
,f x = 1
ont
and0 ≤ f x ≤ 1
. - In a locally compact T2 space
X
, for a compact sett
and a finite family of open sets{s i}
such thatt ⊆ ⋃ i, s i
, there is a family of continuous function{f i}
supported ins i
,∑ i, f i x = 1
ont
and0 ≤ f x ≤ 1
.
These variations are needed in order to prove that rieszContentAux is indeed a Borel measure (following Rudin "Real and complex analysis").
- [x] depends on: #12459
The part using prod_mem
is suggested by @eric-wieser.