Embeddable spaces and duality in topological categories
It was proved in [HM] that each topological group (G,·,τ) may be embedded into a connected topological group (Ĝ,•,τ̂). In fact, two methods of introducing τ̂ were given. In this note we show relations between them.
A space is discretely absolutely star-Lindelöf if for every open cover of and every dense subset of , there exists a countable subset of such that is discrete closed in and , where . We show that every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed subspace.
In this paper we address the following question due to Marcy Barge: For what f:I → I is it the case that the inverse limit of I with single bonding map f can be embedded in the plane so that the shift homeomorphism extends to a diffeomorphism ([BB, Problem 1.5], [BK, Problem 3])? This question could also be phrased as follows: Given a map f:I → I, find a diffeomorphism so that F restricted to its full attracting set, , is topologically conjugate to . In this situation, we say that the inverse...
We prove that every planar rational compactum with rim-type ≤ α, where α is a countable ordinal greater than 0, can be topologically embedded into a planar rational (locally connected) continuum with rim-type ≤ α.
We prove in particular that a continuous image of a Valdivia compact space is Corson provided it contains no homeomorphic copy of the ordinal segment . This generalizes a result of R. Deville and G. Godefroy who proved it for Valdivia compact spaces. We give also a refinement of their result which yields a pointwise version of retractions on a Valdivia compact space.
It is shown how to embed the polydisk algebras (finite and infinite ones) into the disk algebra A(𝔻̅). As a consequence, one obtains uniform closed subalgebras of A(𝔻̅) which have arbitrarily prescribed stable ranks.