Sufficient conditions are given for the union of two Hilbert cube (manifolds) intersecting in a Hilbert cube (manifold) to be a Hilbert cube (manifold). The corollaries include a non-stabilized mapping cylinder theorem for embeddings between Hilbert cube manifolds and a sum theorem for Keller cubes.
In this paper it is shown that if X is a connected space which is not pesudocompact, then βX is not movable and does not have metric shape. In particular βX cannot have trivial shape. It is also shown that if X is Lindelöf and KχβX−X is a continuum, then K cannot be movable or have metric shape unless it is a point.
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.
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