On the Stone-Čech compactification of the space of closed sets

Abstract:

For a topological space X, we denote by ${2^X}$ the space of closed subsets of X with the finite topology. If X is normal and ${T_1}$, the map $F \to {\text {cl}_{\beta X}}F$ is an embedding of ${2^X}$ onto a dense subspace of ${2^{\beta X}}$, and, in this way, we regard ${2^{\beta X}}$ as a compactification of ${2^X}$. This paper is motivated by the following question. When can ${2^{\beta X}}$ be identified as the Stone-Čech compactification of ${2^X}$? In [11], J. Keesling states that $\beta ({2^X}) = {2^{\beta X}}$ implies ${2^X}$ is pseudocompact. We give a proof of this result and establish the following partial converse. If ${2^X} \times {2^X}$ is pseudocompact, then $\beta ({2^X}) = {2^{\beta X}}$. A corollary of this theorem is that $\beta ({2^X}) = {2^{\beta X}}$ when X is ${\aleph _0}$-bounded.