Exercises 1. Let 7 and J' be two topologies on X. If J' 3 7. what does connectedness of X in one topology imply about connectedness in the other? 2. Let [ A. ) be a sequence of connected subspaces of X. such that A, n Anti # @ for all n. Show that ( J A,, is connected. 3. Let [Aa] be a collection of connected subspaces of X; let A be a connected subspace of X. Show that if AnA, # @ for all a, then AU(( A.) is connected. 4. Show that if X is an infinite set. it is connected in the finite complement topology. 5. A space is totally disconnected if its only connected subspaces are one-point sets. Show that if X has the discrete topology, then X is totally disconnected. Does the converse hold? 6. Let A C X. Show that if C is a connected subspace of X that intersects both A and X - A. then C intersects Bd A. 7. Is the space Re connected? Justify your answer. 8. Determine whether or not R" is connected in the uniform topology. 9. Let A be a proper subset of X, and let B be a proper subset of Y. If X and Y are connected, show that ( X x Y) - (A x B) is connected. 10. Let (Xaloe) be an indexed family of connected spaces; let X be the product space X = [X.- Let a = (a) be a fixed point of X. (a) Given any finite subset K of J. let Xx denote the subspace of X consisting of all points x = (X) such that Ja = 0. for a # K. Show that Xx is connected. (b) Show that the union Y of the spaces Xx is connected. (c) Show that X equals the closure of Y': conclude that X is connected. 11. Let p : X - Y be a quotient map. Show that if each set p-([y)) is connected. and if Y is connected, then X is connected. 12. Let Y C X; let X and Y be connected. Show that if A and B form a separation of X - Y, then Y U A and Y U B are connected