Re: Definitions of T_0, T_1, etc. Topological Spaces

Actually, I don't think the term "Frechet space" is used much in topology anyway because of the confusion with "Frechet space" in functional analysis.

I did see in Wikipedia (The article "History of the Separation Axioms" http://en.wikipedia.org/wiki/History_of_the_separation_axioms) that the differences in terminology that I discussed apparently happened because topologists working on metrizability problems often assumed their spaces were T_1 (since metric spaces are T_1) but topologists working in more general spaces often did not assume T_1 to make their results as general as possible. But also there apparently were some topologists who did assume T_1 for T_3, T_3.5, etc. so that the chain T_6 implies T_5, T_5 implies T_4, etc. is preserved.

And actually Hocking and Young's book and the "Counterexamples in Topology" book have both been around for decades, so these books might not use the same terminology as modern topologists use it. That Wikipedia article did mention the uses of these words have changed over time and I think it did suggest that perhaps the uses of these terms have changed (perhaps becoming more universal) since these two books have been written. And I wonder how consisent Munkres' use of the terminology is with other modern topologists' use of these terms.
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