Topology, Volume 1

0-dimensional A₁ additive class ambiguous of class ambiguous sets analytic sets arbitrary assume Axiom B₁ Baire property belongs Borel set boundary set cartesian product class F closed and open closed sets compact complete space condition Consequently contains continuous function continuous mapping countable set defined definition denote dense set dimension disjoint ensembles equivalent exists a sequence f is continuous F-set F₁ finite number following theorem formula Fr(X function f functions of class Fund G is open G-set G₁ G₂ H₁ Hausdorff hence homeomorphism hypothesis implies infinite intersection k₁ Let f Math metric space multiplicative class neighbourhood non-empty one-to-one open set G open sets propositional function prove real numbers Remark satisfying Section separable space sets of class Sierpiński simplex Suppose T1-space topological space transfinite uncountable union Urysohn whence X₁ Y₁