Extension closed and cluster closed subspaces

Douglas Harris
1972 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Introduction. One of the most useful properties of a compact Hausdorff space is that such a space is closed whenever embedded into a Hausdorff space. This property does not extend to compact spaces with respect to embeddings into arbitrary spaces. Thus, an interesting topological problem is to characterize the types of absolute "closure" properties that are possessed by compact spaces. This is the problem that is solved in the present paper. The following notation and terminology will be used
more » ... low. We shall consider a fixed space X and subspace A, representing arbitrary nonempty open subsets of X (respectively A ) by W (respectively V). Collections of nonempty open subsets of X (respectively A) are denoted by x (respectively a) and an extension of a to X is a collection % whose trace y\A = {WC\ A : W G x} (with empty intersections deleted) is equal to a; such extensions always exist. A cover of X (respectively A) is a collection x (respectively a) whose union is X (respectively A); a, cover is infinite if it has no finite subcover. For each V define V x to be the union of all W such that W C\ A C V, and for each a define a x = { V x : V G a}. A filterbase X on A converges (clusters) The proofs of the results given in this paper are straightforward, and are omitted (except for that of Proposition A). Extension closed subspaces. Among Hausdorff spaces, the closed subspaces are characterized by the property that such a subspace is extension closed] that is, every cover of the subspace extends to a cover of the entire space. A number of equivalent characterizations of extension closed subspaces can be given. A few preliminary properties must be set forth.
doi:10.4153/cjm-1972-119-8 fatcat:ypczmm7o5rczbmhtspjyuecox4