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.
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
$\mathcal{T}^*$ is coarser than $\mathcal{T}$, i.e., $\mathcal{T}^* \subseteq \mathcal{T}$;
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.
A topological space $X$ is called cocompact if there is a cotopology $\mathcal{T}^*$ on $X$ which is compact.
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.
The concept principally appeared in
De Groot made cotopologies popular by giving a unifying and generalizing version of the Baire category theorem
unfortunately, his proof contained a gap that was later closed by
In this context some
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.
Last revised on March 18, 2019 at 09:26:36. See the history of this page for a list of all contributions to it.