Abstract
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:
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