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absolute value metric absolute value topology basis Cauchy sequence Chapter closed subset coﬁnally compact space compact T2-space complete metric space connected subspace continuous function converges Corollary countable family countably compact deﬁned denote directed set discrete topology disjoint element equivalence classes equivalence relation Example 11 Exercise f is continuous ﬁlter Q ﬁnd ﬁner ﬁrst countable following deﬁnition function f fundamental group given hence Hilbert cube homeomorphic intersection limit point locally compact locally connected nonempty subsets one-one open cover open neighborhood system open sets open subset partially ordered positive integers positive number product space product topology Proposition 11 Pythagorean metric rational numbers reader real numbers residually satisﬁes second countable set of real set which contains space of real subbasis subset of R2 subspace topology Suppose f topological space topology induced trivial topology ultraﬁlter ultranet