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feat(Topology/UrysohnsLemma): add variations of Urysohn's lemma

Open yoh-tanimoto opened this issue 10 months ago • 3 comments

add variations of Urysohn's lemma:

  • In a normal space X, for a closed set t and an open set s such that t ⊆ s, there is a continuous function f supported in s, f x = 1 on t and 0 ≤ f x ≤ 1.
  • In a locally compact T2 space X, for a compact set t and a finite family of open sets {s i} such that t ⊆ ⋃ i, s i, there is a family of continuous function {f i} supported in s i, ∑ i, f i x = 1 on t and 0 ≤ 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.

yoh-tanimoto avatar Apr 19 '24 12:04 yoh-tanimoto