Compact-open topology

2020 Mathematics Subject Classification: Primary: 54C35 [MSN][ZBL]

One of the topologies on the set of mappings of one topological space into another. Let $F$ be some set of mappings of a topological space $X$ into a topological space $Y$. Each finite collection of pairs $(X_1,U_1),\ldots,(X_n,U_n)$, where $X_i$ is a compact subset of $X$ and $U_i$ is an open subset of $Y$, $i=1,\ldots,n$, determines the subset of mappings $f \in F$ for which, for all $i$, $f[X_i] \subseteq U_i$; the family of all such sets is the base for the compact-open topology on $F$. The importance of compact-open topologies is due to the fact that they are essential elements in Pontryagin's theory of duality of locally compact commutative groups and participate in the construction of skew products. If $Y$ is a Hausdorff space, the compact-open topology also satisfies the Hausdorff separation axiom. If all mappings $f \in F$ are continuous and if $Y$ is a completely-regular space, then $F$ endowed with the compact-open topology is completely regular. On the assumption that all mappings $f \in F$ are continuous and that $X$ is a locally compact Hausdorff space, the compact-open topology on $F$ is admissible or compatible with continuity, i.e. the mapping $\phi : F \times X \rightarrow Y$ defined by the formula $\phi : (f,x) \mapsto f(x)$ is continuous, and the compact-open topology is the smallest (weakest) of all topologies on $F$ for which $\phi$ is continuous. In this respect the compact-open topology is preferable to the topology of pointwise convergence, since the latter is usually weaker than the former, and is not admissible in such a case. Moreover, the fact that the group of homeomorphisms of a Hausdorff compactum $K$ into itself, endowed with the compact-open topology, is a topological group which acts continuously (in the above sense) on $K$, is of fundamental importance. The group of homeomorphisms of an arbitrary locally compact Hausdorff space into itself need not be a topological group with respect to the compact-open topology (the transition to the inverse element may prove not to be a continuous mapping with respect to this topology), but if a locally compact Hausdorff space $K$ is locally connected, the compact-open topology again converts the group of all homeomorphisms of $K$ into itself into a topological group acting continuously on $K$. This is an important result, since all manifolds are locally compact and locally connected.

References

[1]	J.L. Kelley, "General topology" , Springer (1975)
[2]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[3]	N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)
[4]	R.F. Arens, "Topologies for homeomorphism groups" Amer. J. Math. , 68 : 4 (1946) pp. 593–610