This group gathers theorems from general topology, also called point-set topology, the study of open sets, continuity, compactness, and separation in abstract topological spaces. It covers Tychonoff's theorem that any product of compact spaces is compact, Urysohn's lemma and the Tietze extension theorem on separating closed sets and extending continuous functions, the metrization theorems of Urysohn and of Nagata and Smirnov that say when a space can be given a metric, and Cantor's intersection theorem on nested compact sets.
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Tychonoff's Theorem (Wikipedia)
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In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology.
Urysohn's Lemma (Wikipedia)
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In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.
Tietze Extension Theorem (Wikipedia)
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In topology, the Tietze extension theorem (also known as the Tietze-Urysohn-Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
Nagata-Smirnov metrization theorem, Wikipedia
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In topology, the Nagata-Smirnov metrization theorem characterizes when a topological space is metrizable.
Cantor's Intersection Theorem (Wikipedia)
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Cantor's intersection theorem, also called Cantor's nested intervals theorem, refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets.
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