= First-countable space
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= First countable
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= First countability
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A <topological space> is first-countable when each point has a countable <neighbourhood basis>. Every <metric space> is first-countable: the <open balls> $B(x,1/n)$ form such a basis. A subset of a first-countable space remains first-countable in its <subspace topology>, since intersecting a local basis with the subset preserves the defining containment property. Thus a space failing first countability cannot be <homeomorphic> to any <subset> of a <metric space>.
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