Connected topological space, DYK Fact #167

[JaneFairfax]

Given a topological space X, the empty set and X itself are always “clopen” sets – sets that are both open and closed. (Note: We say that a subset of the topological space X is closed iff its complement in X is open.) If X has no other “clopen” sets, it is said to be connected.

On the other hand, a topological space is said to be disconnected iff it is the disjoint union of two nonempty open sets.

If a topological space Y is disconnected, then Y is the disjoint union of nonempty open sets S and T. Then S and T are closed (since they are each other’s complement), and they are also proper subsets of Y (since neither of them is empty). Hence Y, having “clopen” subsets other than the empty set and itself, is not connected. Conversely, if Y is not connected, then there exists a clopen set S which is neither the empty set nor Y. Since S is closed, its complement in Y is open, and since S is not the whole of Y, its complement in Y is not empty. Hence Y, being the disjoint union of S and its complement in Y, both of which are open and non-empty, is disconnected.

Did you know? This proves that a topological space is disconnected if and only if it is not connected.

[algebraic topology]

Did you know?

A topological space X is connected if and only if there exists no surjective continuous function from X onto the discrete two-element space.

[JaneFairfax]

Did you know? A topological space is locally connected iff any element of any open subset belongs to a connected open set contained in that open subset. 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.

[algebraic topology]

Did you know?

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.

[JaneFairfax]

Did you know? A more obvious – but far less interesting – covariant functor from the category of topological spaces and continuous functions to the category of sets and set functions is the one that maps each topological space to its underlying set. Such a functor is said to be forgetful because it “forgets” some of the structures of the first category!

[algebraic topology]

Did you know?

The opposite of a forget functor is a free functor, e.g. the functor from the category of sets and set functions to the category of groups and group homomorphisms mapping each set to the free group generated by that set.