### A note on strong Jordan separation.

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Knot complements in the n-sphere are characterized. A connected open subset W of ${S}^{n}$ is homeomorphic with the complement of a locally flat (n-2)-sphere in ${S}^{n}$, n ≥ 4, if and only if the first homology group of W is infinite cyclic, W has one end, and the homotopy groups of the end of W are isomorphic to those of ${S}^{1}$ in dimensions less than n/2. This result generalizes earlier theorems of Daverman, Liem, and Liem and Venema.

In this survey, we consider several questions pertaining to homeomorphisms, including criteria for their existence in certain circumstances, and obstructions to their existence.

In each manifold $M$ modeled on a finite or infinite dimensional cube ${[0,1]}^{n}$, $n\le \omega $, we construct a meager ${F}_{\sigma}$-subset $X\subset M$ which is universal meager in the sense that for each meager subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h\left(A\right)\subset X$. We also prove that any two universal meager ${F}_{\sigma}$-sets in $M$ are ambiently homeomorphic.