Youtor
Hausdorff Example 2 Quotient Space
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=fymrnh9Emn8
EDIT: (3:46) In general, a quotient map q need not be open (carry open set to open sets). If we check in this example (next board), it turns out to be open. For the open sets in the quotient, the usual topology for the circles should be clear: inverse images of small open intervals are unions of open intervals. On the other hand, if a subset of a circle has open inverse image, then each inverse point is contained in an open interval, small enough to map back bijectively. Thus the subset is a union of open subsets. If an open subset contains q(0), the inverse image must contain an open interval containing zero, so everything. • Point Set Topology: Let X be the real line and consider the equivalence relation: xRy iff x and y differ by a multiple of 2^k (k an integer). We show that the quotient space associated to R is not Hausdorff.
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