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Unformatted text preview: MATH 202A HOMEWORK 6 This assignment is due Wed., Oct. 10. The topics addressed are connectedness, products and quotients. Problem 1 . (a) Let X be a topological space, { C α } be a collection of connected subsets such that any two of them have a point in common. Prove that G = [ C α is connected. [It follows that our definition of connected component containing x is connected, and is the largest connected set containing x .] (b) Let X be a topological space and Y ⊂ X . Prove that if Y is connected, then ¯ Y is connected. [It follows that the connected components of any topological space are closed.] (c) Prove that the components of a locally connected topological space are open. (d) Prove that the topologists sine curve is not path connected. [In class, we proved that it is connected.] (e) Find an example of a connected space with ≥ 3 path connected components. (f) Prove that the Cantor set is totally disconnected. (g) Prove that R 2 \ Q 2 is connected....
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 Spring '10
 tao/analysis
 Logic, Topology, Sets, Topological space, α∈I Xα, connected spaces

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