Question

Problem 24.8:(a,b,c). (c) If tAn is a collection of path-connected subspaces of X and if N-,メ0, is UA, necessarily path connected? Solution: (c) Yes. Let Y1,Y2 E Y and choose x1,x2 E Y 2 E X such that f(x) = yi and /(22) = y. If p : [0.1] → X is a path from xi to T2 in X, then fop : 10.11 → y is a path from yi to y2 in and choose r.

what is the question saying. if the intersection of the collection is an emptyset. why is it that the big union of the collection is path connected. what is meant by path connected here. what is the solution suggesting? Can somebody please explain the question and solution?

Answer 1

any doubt ask in comment

Path connected A set x is said to be path connected. if x_2, x_2 epsilon x then there exist a continuous (path) function f: [a, b] rightarrow x such that f(Q) = x_1 and f(b) = x_2 for some a, b epsilon IR a < b {A_alpha} collection of path connected. subspace of x if intersection of collection is empty take example simply x = IR, real number with usual A_1 = [0, 1] A_2 = [2,3] so A_1 U A_2 = [0, 1] U[2, 3] which is not path connected (even not connected) now the given collection with given information we show UA_alpha is path connected using definition take x_1, x_2 epsilon U A_alpha implies x_1 epsilon A_alpha_1 x_2 epsilon A_alpha_2 for some alpha_1 alpha_2 index set continuous also let y epsilon Intersection A_alpha implies y epsilon A_alpha_i and y epsilon A_alpha_2. exist f_1: [a, b] rightarrow A_alpha_2 such that f_1 (a) = x_1 f_2(d) = x_2

Take X =R

And take finite collection of intervals of R with empty and non empty intersection for better understanding .