On the Stone-Cech Compactification of the Space of Closed Sets

John Ginsburg
1976 Transactions of the American Mathematical Society  
For a topological space X, we denote by 2 the space of closed subsets of X with the finite topology. If X is normal and Tx, the map F -► cl« VF is an embedding of 2 onto a dense subspace of 2^ , and, in this P fíY X way, we regard 2P as a compactification of 2 . This paper is motivated by QX v the following question. When can 2H be identified as the Stone-Cech comy Y QX X pacification of 2 ? In [11], J. Keesling states that 0(2 ) = 2* implies 2 is pseudocompact. We give a proof of this result
more » ... d establish the following XX X QX partial converse. If 2 x 2 is pseudocompact, then (3(2 ) = 2V . A corollary of this theorem is that 0(2 ) = 2^ when X is Nn-bounded.
doi:10.2307/1999729 fatcat:mxnn6vjtwrfdjnbzray4vrlnve