Open-problem statements, with their sources attached.

The four dimensional case of the smooth version of the conjecture is still open. See [Wang2017].

It is conjectured that the only values of $n > 4$ for which the smooth version of the conjecture holds are $n = 5, 6, 12, 56, 61$. See Conjecture 1.17 in [Wang2017].

Is there a Lindelöf space with singletons as Gδ sets with cardinality greater than the continuum?

Assume for $n>1$, $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology. Does the connectedness of (the induced power set map) $f$ imply that of $f^{-1}$?

Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?

Problem 14 in [Ar2013]: Is it possible to represent an arbitrary compact hausdorff space as an image of a homogeneous compact space under a continuous mapping?

Problem 15 in [Ar2013]: Is every homogeneous ω-monolithic compact hausdorff space first countable?

Problem 16 in [Ar2013]: Is the cardinality of every homogeneous ω-monolithic compact hausdorff space not greater than 𝔠?

Problem 17 in [Ar2013]: Is it true that every nonempty ω-monolithic compact hausdorff space contains a point with a first countable neighborhood basis?

Note: Nonempty X is required since the conclusion asserts the existence of a point.

Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space X such that $𝔠 < |X|$.

Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.

Problem 3 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space which is not first countable.

Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.

The Bing-Borsuk Conjecture: every $n$-dimensional homogeneous absolute neighborhood retract is a topological $n$-manifold. A topological space $X$ is an $n$-dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X). The hypothesis [MetrizableSpace X] implies T2Space X so this does not appear in the conclusion.

Any T2, Toronto space is discrete.