Math Is Fun Forum

A topological space is locally connected iff given any open subset U and any x ∈ U, there exists a connected open set V such that x ∈ V ⊆ U. A subset of a topological space is a maximal connected subset iff it is connected and is not properly contained in any other connected subset. Any maximal connected subset of a locally connected topological space is open, and any topological space is the disjoint union of its maximal connected subsets. The cardinality of the class of all maximal connected subsets of a topological space is a topological invariant, meaning that it is preserved under homeomorphisms. (In the language of category theory, there is a covariant functor from the category of topological spaces and continuous fuctions to the category of sets and set functions which maps each topological space to the set of its maximal connected subsets, with homeomorphisms in the first category corresponding to bijections in the other.)

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