It is shown that, for a Tychonoff space X, the complete upper semilattice K(X) of compactifications of X is a lattice if either (1) βX⧹X is realcompact and C∗-embedded in βX, or (2) βX⧹X is a P-space and clβX(βX⧹X) is an F-space. The concept of bounding lattice is introduced and examples of spaces X are given such that K(X) is a lattice but not a bounding lattice. A certain class of Tychonoff spaces X is constructed such that K(X) is a lattice.
The set theoretic principle ♢ is used to construct hereditarily Lindelof, non-separable subspaces of given complete spaces of countable tightness. The construction is patterned after R. B. Jensen's original use of ♢ to construct a Souslin line, and yields the following result: Suppose X is a regular space of countable tightness having weight at most c. If no non-empty Gδ set in X is contained in a separable subspace of X, and if either X is countably complete or has all closed subsets Baire, then X contains an L-space.
Finite dimensional techniques of Bing and Bryant are extended to Hilbert cube manifolds to show that where M is a Hilbert cube manifold, A is an embedded copy of 1k, 0k∞, and Q is the Hilbert cube. Among the corollaries given here are elementary proofs of two theorems of West: the mapping cylinder theorem and the sum theorem for Hilbert cube factors.
We show that a known restriction on the cardinalities of closures of subspaces of scattered We then find a wide class of spaces, , cannot be improved to , for any λ.T.3. scattered spaces which have no scattered compactification: these spaces are derived from regular filters over cardinals bigger than N1.
We say that a subset S of a topological space X is M-embedded (MN0-embedded) in X if every map from S to a (separable) metrizable AE can be extended over X. Characterizations of M-and MNO-embedding are given and we prove that S is M-embedded (MNO-embedded) in X iff(X,S) has the Homotopy Extension Property with respect to every (seperable) ANR space.
One of the aims of this paper is to generalize a theorem of P. Fletcher and W.F. Lindgren characterizing second countable spaces. On behalf of that, we investigate a basis property related to the concept of σ-Q-bases defined by Fletcher and Lindgren and orthobases studied by P. Nyikos and W.F. Lindgren. In this new setting we state a necessary and sufficient condition for ωμ-quasimetrizability of topological spaces and we discuss a problem of P. Fletcher and W.F. Lindgren and a related theorem of S. Nedev concerning quasimetrizability of T1-spaces. As a corollary we give a characterization of ωμ-additive spaces having a base of cardinality ωμ— In the second part of the paper, we study (not necessarily symmetric) distance-functions on a space X taking their values in a partially ordered group H. We show that every T1-space X is quasimetrizable in this generalized sense.
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：