Could you explain how to show 8.11 in detail? Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each ofX and Y is connected.Definition. Let X be a topological space. Then X is connected if and only if X is notthe union of two disjoint non-empty open sets.Definition. Let X be a topological space. Subsets A, B in X are separated if and onlyif An B = A n B = Ø. Thus B does not contain any limit points of A, and A does notcontain any limit points of B. The notation Xare separated sets.A | B means X = A U B and A and BTheorem 8.1. The following are equivalent:(1) X is connected.(2) There is no continuous function f : X → Rstd such that f(X) = {0, 1}.(3) X is not the union of two disjoint non-empty separated sets.(4) X is not the union of two disjoint non-empty closed sets.(5) The only subsets of X that are both closed and open in X are the empty set and X itself.(6) For every pair of points p and q and every open cover {Uq}aea Qf X there exist a finitenumber of the Ua's, {U«,, U«,, Uaz,.., Ua, } such that p E Ug,, q E U,, and for eachi < n, Ua, n Uau1 # Ø.Theorem 8.3. The space Rstd is connected.Theorem 8.5. Let {Ca}aea be a collection of connected subsets of X, and let E be anotherconnected subset of X such that for each a in 1, E n Ca # Ø. Then E U (Ure, Ca) isαελconnected.Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset ofX such that C cDC C, then D is connected.Theorem 8.9. Let f : X→ Y be a continuous, surjective function. If X is connected,then Y is connected.Theorem 8.10 (Intermediate Value Theorem). Let f : Rstdmap. If a, b e R and r is a point of R such that f(a) < r < f(b), then there exists a pointc in (a, b) such that f(c) =Rstd be a continuous= r.

Name: Could you explain how to show 8.11 in detail? Theorem 8.11. For topolo
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Could you explain how to show 8.11 in detail? Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each ofX and Y is connected.Definition. Let X be a topological space. Then X is connected if and only if X is notthe union of