hw5

MATH 4181 001 Fall 1999 Problem Set 5 1. Letf:X!Ybe a function. Then fis an open mapping if for each open set OX,f(O) is open in Y. (a) Give an example of a mapping that is continous, but not open. (b) Give an example of a mapping that is open, but not continous. (c) Prove that a one-to-one, onto mapping f:X!Yis a homeomorphism if and only iffandf1are open mappings. 2. Prove that a nite subset Aof a Hausdor space Xhas no limit points. Conclude that Amust be closed. 3. IfXis a space which is homeomorphic to a subspace Aof a spaceY, thenXis said to beembedded inY. Give an example of spaces AandBfor whichAcan be embedded inBandBcan be embedded in A, butAandBare not homeomorphic. (Simple examples can be found in R.) 4. Prove that every countable subset of Ris totally disconnected. 5. Give examples of subsets AandBinR2to illustrate each of the following. A drawing is sucient. (a)AandBare connected, but A\Bis disconnected. (b)AandBare connected, but AnBis disconnected. (c)AandBare disconnected, but A[Bis connected. (d)AandBare connected and A\B6=;, butA[Bis disconnected. 6.Denition: A Hausdor space Xis0-dimensional ifXhas a basis Bof sets which are simultaneously open and closed. Prove that every 0-dimensional space is totally disconnected. 7. Prove: (a) The property of being totally disconnected is a topological invariant but not a continuous invariant. (b) The property of being totally disconnected is hereditary. 1