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Working with Topological Spaces

Which of the following topological spaces is normal?

a) Reals with the "usual topology."
b) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X.
c) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X.
d) Reals with the "lower limit topology:" basis half-closed intervals [a,b)
e) Reals with the "upper limit topology:" basis half-closed intervals (a,b]
f) Reals with the "K-topology:" basis consists of open intervals (a,b) and sets of form (a,b) - K where K = {1, 1/2, 1/3, ... }

Solution Preview

We know, a normal topological space is a T4 space. For any closed non-intersection sets A,B in X, there exists open neighborhoods of U(A) and U(B) such that U(A) intersects U(B) is empty.

In the following problems, I think you mean X represents the set of reals.
a) Yes. Any metric space is normal.

b) No. Suppose A={1}, B={2}, by definition, X-A and X-B are open sets since ...

Solution Summary

This solution discusses in about 290 words, six different problems and uses proof to illustrate which display normal topological spaces.

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