Question

want proof for theorem 7.12 using definition 7.9

Problem 7.7. Give an example of a space that is connected, but not path con- nected. Problem 7.8. Show that R" is not homeomorphic to R if n>1 Definition 7.9. Let be a point in X. Then X is called locally path connected at a if for each open set U containing r, there is a path connected open set V containing r such that V CU. If X is locally path connected at every point, then X is path connected Theorem 7.10. The following are equivalent: (1) X is locally path connected. (2) IfU is open in X, then every path component of U is open in X Theorem 7.11. If X is connected and locally path connected, then X is path connected. Problem 7.12. Give an example of a locally path connected space X and a con- tinuous function f from X to a topological space Y so that f(X) is not locally path Definition 7.13. X is a Baire space if the intersection of each countable family of dense open sets is dense. A set A CX is nowhere dense in X if (A). A set ACX is first category in X if A-UR1 An, where each An is nowhere dense in X. If a set is not first category, is called second category. (Topologically second category sets in X are "thick" and first category sets in X are "thin") Type here to search 3:31 PM 3/9/2019

Answer 1

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