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Kevin_H
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The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom.
We are going over separation axioms in class when we were asked to prove that every Urysohn Space is a Hausdorff.
What I understand:
We are going over separation axioms in class when we were asked to prove that every Urysohn Space is a Hausdorff.
What I understand:
- A space ##X## is Urysohn space provided whenever for any two distinct points of ##X## there are neighborhoods of ##U## of ##x## and ##V## of ##y## such that ##\overline{U}\cap \overline{V}=\emptyset.##
- A space ##X## is Hausdorff if for any two distinct points of ##X## can be separated by open neighborhoods ##(x\in U,y\in V,U\cap V=\emptyset)##.
- Given ##(X,\mathscr{T})## to be a Urysohn space, let ##x,y\in X## be distinct points. Then there are neighborhoods of ##U## of ##x## and ##V## of ##y## such that ##\overline{U}\cap \overline{V}=\emptyset##. We seek to prove there exists ##x\in B## open ##\subset X## and ##y\in A## open ##\subset X## such that ##A\cap B=\emptyset##. Consider ##X\setminus\overline{U}##. This is an open neighborhood of ##y##, thus there exists ##y\in A\subset\overline{A}\subset X\setminus\overline{U}##. Consider ##X\setminus \overline{A}##. This is an open neighborhood of ##x##, thus there exist ##x\in B\subset X\setminus\overline{A}.## Thus ##A\cap B=\emptyset##. QED.