Sesión 411 (About the space of continuous functions with open domain)

Agosto 26 de 2024 (Sesión 411)

About the space of continuous functions with open domain

Functions with open domains appear in a variety of contexts in mathematics. Motivated by this phenomenon, M. Allah and R. Brown set out to endow the space of continuous functions with open domains with an appropriate topology. In chronological order, in [1], they define the topology $\tau_{co}$, which turns the inverse semigroup of partial homeomorphisms between open sets into a topological inverse semigroup, called $\Gamma(X)$ for $X$ locally compact Hausdorff space. More recently, in [3], they added open sets after realizing that this topology is not $T_1$, showing that this addition makes $\Gamma(X)$ a Hausdorff topological inverse semigroup. Then, in [2], they show that if $X$ is a compact, Hausdorff, and totally disconnected space, then $(\Gamma(X),\tau_{hco})$ is completely metrizable. In what follows, we will see how to define such a topology, we will see that the space of continuous functions with open domains $(Cod(X,Y),\beta)$ is a Polish space when $X$ is locally compact, Hausdorff and second countable space and $Y$ is a complete space. In particular, we will find a metric for $(\Gamma(X),\tau_{hco})$.

References

[1] A. Allah and R. Brown. A compact-open topology on partial maps with open domain. J. London Math. Soc. (2), 21(3):480-486, 1980.

[2] J. Perez, C. Uzcátegui. On the Polishness of the inverse semigroup Γ(X) on a compact metric space X. European Journal of Mathematics (2023) 9:113.

[3] L. Martínez, H. Pinedo and C. Uzcátegui. A topological correspondence between partial actions of groups and inverse semigroup actions. Forum Math., 34(2), 431-446, 2022.

Expositor: Edwar Ramirez (joined work with Carlos Uzcátegui).