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June 30th, 2012, 07:36 AM   #1
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Uniform Spaces: bases and subbases for uniformities

Prove that if 'curly E' is a base for the uniformity 'curly D' consisting of open sets, then {E[x]: E in 'curly E', x in X} is a base for the topology of X.

So what I have to show is each E[x] is open and that every open set contains a member of {E[x]:E in 'curly E', x in X}. The 2nd part is easy, I want to ask about the first part, since E is open in X x X does it follow immediately that E[x] is open? If so, how? If not can you assist with a proof of the first requirement for a base of the topology mentioned above.

By the way this is a question from the book 'General Topology' by Stephen Willard page 243 problem35D. Thanks.
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