WebDefinition. There are several competing definitions of a "proper function".Some authors call a function : between two topological spaces proper if the preimage of every compact set in … WebThe answer is negative even for ${\rm{PGL}}_2$: the stabilizer of an edge in the building is a counterexample (with Iwahori preimage in ${\rm{SL}}_2(F)$).
Math 341 Lecture #22 x4.4: Continuous Functions on Compact Sets
WebAug 12, 2024 · Inverse image of compact is compact. Let f: X → Y be a closed map of topological spaces, such that the inverse image of each point in Y is a compact subset of … Web5. Locally compact spaces Definition. A locally compact space is a Hausdorff topological space with the property (lc) Every point has a compact neighborhood. One key feature of locally compact spaces is contained in the following; Lemma 5.1. Let Xbe a locally compact space, let Kbe a compact set in X, and let Dbe an open subset, with K⊂ D. toys are us tokyo
algebraic groups - Preimage of a maximal compact open subgroup in …
Web4. In class, we proved that the continuous image of a compact set is compact and the continuous image of a connected set is connected. What about the preimages? More precisely, let f: R → R be a continuous function. Prove or disprove the following: - If K ⊂ R is compact, then the preimage f −1[K] = {x ∈ R ∣ f (x) ∈ K } is compact. WebAug 30, 2024 · Let X and Y be Hausdorff spaces and suppose that Y is locally compact. Let f: X → Y be a surjective map such that for any compact subset K ⊂ Y the pre-image. is a compact subset of X. What can I tell about the continuity of f? If Y is compact, then f is certainly continuous if we restrict f to the pre-image of a compact, that is, f f − ... WebOct 23, 2007 · I have seen the word "range" used in two different ways: (1) The target of a function. (R m, for your example) (2) The image of a function. (f (R n ), in your example) It … toys are us weekly ad