[Merged by Bors] - feat(topology/instances/nnreal): generalize has_continuous_smul instance
#16713
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…tance (#16713) This generalizes the `has_continuous_smul ℝ≥0 ℝ` instance to `has_continuous_smul ℝ≥0 α` whenever there is a `mul_action ℝ α` with `has_continuous_smul R α` (the `mul_action` is the minimal assumption to get an induced action of `ℝ≥0` on `α` in mathlib).
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has_continuous_smul instancehas_continuous_smul instance
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…spondence between closed ideals in `C(X, 𝕜)` and open sets in `X` (#16677) For a topological ring `R` and a topological space `X` there is a Galois connection between `ideal C(X, R)` and `set X` given by sending each `I : ideal C(X, R)` to `{x : X | ∀ f ∈ I, f x = 0}ᶜ` and mapping `s : set X` to the ideal with carrier `{f : C(X, R) | ∀ x ∈ sᶜ, f x = 0}`, and we call these maps `continuous_map.set_of_ideal` and `continuous_map.ideal_of_set`. As long as `R` is Hausdorff, `continuous_map.set_of_ideal I` is open, and if, in addition, `X` is locally compact, then `continuous_map.set_of_ideal s` is closed. When `R = 𝕜` with `is_R_or_C 𝕜` and `X` is compact Hausdorff, then this Galois connection can be improved to a true Galois correspondence (i.e., order isomorphism) between the type `opens X` and the subtype of closed ideals of `C(X, R)`. As a consequence, the maximal ideals correspond precisely to (complements of) singletons, but this fact will appear in a separate PR. - [x] depends on: #16664 - [x] depends on: #16709 - [x] depends on: #16713 - [x] depends on: #16714
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…to (complements of) singletons (#16719) This establishes the correspondence between maximal ideals of `C(X, 𝕜)` (where `X` is compact Hausdorff and `is_R_or_C 𝕜`) and the complements of singletons in `X`. This allows for the proof that the natural map from `X` to `character_space 𝕜 C(X, 𝕜)` is a homeomorphism. - [x] depends on: #16709 - [x] depends on: #16713 - [x] depends on: #16714 - [x] depends on: #16677 - [x] depends on: #16663 - [x] depends on: #16721 - [x] depends on: #16722 - [x] depends on: #16801
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This generalizes the
has_continuous_smul ℝ≥0 ℝinstance tohas_continuous_smul ℝ≥0 αwhenever there is amul_action ℝ αwithhas_continuous_smul R α(themul_actionis the minimal assumption to get an induced action ofℝ≥0onαin mathlib).