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On spaces which are essentially T1

Published online by Cambridge University Press:  09 April 2009

Dick Wick Hall ,
Sheila K. Murphy  and
Eugene P. Rozycki
Dick Wick Hall
Affiliation:
State University of New York at Binghamton
Sheila K. Murphy
Affiliation:
State University of New York at Binghamton
Eugene P. Rozycki
Affiliation:
State University of New York at Binghamton
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Let S be a pseudo-metric space with pseudo-metric d. Then for each nonempty A ⊆ S, x ∈ cl (A) if and only if d(x, A) = 0, and d(x, y) = d(y, x) for all x, y ∈S. Thus, if d(x, y) = 0, then x∈cl({Y}) by the first requirement and Y∈cl({x}) by the second requirement. It is natural then to expect that if a topological space S is to be pseudo-metrizable, it should at least satisfy the requirement:

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 4 , November 1971 , pp. 451 - 455
Copyright
Copyright © Australian Mathematical Society 1971

References

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