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.
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.