nLab cotopology

Idea

Formally, a cotopology of a topological space $(X, \mathcal{T})$ is a coarser topology $\mathcal{T}^*$ on the same set $X$ that does not forget too many sets. It can be used to generalize the Baire category theorem and to characterize topological completeness. The pair of both topologies, the original and the coarser one, constitute an example of a bitopological space.

Definitions

Definition

Let $(X, \mathcal{T})$ be a topological space. A topology $\mathcal{T}^*$ on $X$ is called a cotopology of $\mathcal{T}$—and $(X, \mathcal{T}^*)$ is called a cospace of $(X, \mathcal{T})$—if

1. $\mathcal{T}^*$ is coarser than $\mathcal{T}$, i.e., $\mathcal{T}^* \subseteq \mathcal{T}$;

2. for each point $x$ and each $\mathcal{T}$-closed $\mathcal{T}$-neighborhood $V$ of $x$ in $X$ there exists a $\mathcal{T}^*$-closed $\mathcal{T}$-neighborhood $U$ of $x$ in $X$ such that $U$ is contained in $V$.

If $\mathcal{T}$ is regular, the last condition can be replaced by other conditions, see this proposition.

Definition

A topological space $X$ is called cocompact if there is a cotopology $\mathcal{T}^*$ on $X$ which is compact.

Properties

Every cocompact regular space is a Baire space. A metrisable space is topologically complete if and only if it is cocompact.

A space that admits only Hausdorff cospaces is equivalently an H-closed space, i.e. it (is Hausdorff and) is not a proper dense subspace of another space.

Related Concepts

• bitopological space

References

The concept principally appeared in

• J. D. Weston, On the comparison of topologies 1956, Journal of the London Mathematical Society, vol. s1-32 no. 3, pp. 342-354.

De Groot made cotopologies popular by giving a unifying and generalizing version of the Baire category theorem

• de Groot, Subcompactness and the Baire category theorem 1963, Nederl. Akad. Wetensch. Proc., Ser. A66 = Indag. Math., vol. 25, pp. 761-767;

unfortunately, his proof contained a gap that was later closed by

• Isidore Fleischer, On ‘Subcompactness and the Baire category theorem’ 1979, Nederl. Akad. Wetensch. Proc. Ser. A 82 = Indag. Math. 41, pp. 9-11.

In this context some

• Colloquium co-topologie at the Mathematisch Centrum, today Centrum Wiskunde & Informatica, in Amsterdam in 1964

is mentioned. Further developments include

• Aarts, de Groot, McDowell, Cotopology for metrizable spaces 1970, Duke Mathematical Journal vol. 37.

• George Strecker and G. Viglino, Cotopology and Minimal Hausdorff Spaces 1969, Proceedings of the American Mathematical Society Vol. 21 No. 3.