General Topology by Willard Stephen;

Author:Willard, Stephen;
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2012-07-17T16:00:00+00:00

for i = 1, . . . , N. Then for n > Nε we find

so that (xn) converges to y, as claimed. ■

We are now ready for the subspace theorem. The first part is due to Alexandroff, the second to Mazurkiewicz. Both are classical results from the 1920’s.

24.12 Theorem. A Gδ-set in a complete space is completely metrizable. Conversely, if a subset A of a metric space M is completely metrizable, it is a Gδ-set.

Proof. First, suppose G is open in the complete space (M, p). Define f(x) = 1/[ρ(x, M - G)] for each x ∈ G. Then f is continuous on G (24E). Now define

ρ*(x, y) = ρ(x, y) + |f(x)–f(y)|

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