General topology

Apply base cardinal number Cartesian product Cech-complete closed mapping closed sets closed subspace closure collectionwise normal compact subspace compact-open topology compactification completely metrizable consider consisting contains continuous function continuous mapping convergence Corollary countably compact countably paracompact Deduce denoted diagonal discrete space Exercise exists first-countable functionally closed functionally open Give an example Hausdorff space Hence Hint homeomorphic homeomorphic embedding implies invariant inverse system Lemma Lindelof space linearly ordered locally compact space mapping f metric Q metrizable space natural number neighbourhood non-empty normal space one-point open cover open mapping open refinement open sets open subsets pair paracompact space perfect mapping perfectly normal Problem Proof Proposition Prove proximity pseudocompact pseudometric quotient mapping readily verify real line realcompact regular space Section separable sequential space space X suffices to show T0-space topological space topology induced totally bounded Tychonoff space uniform space uniformly continuous Urysohn virtue of Theorem