2 problems

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Borsuk's finite-polyhedron conjecture for compact metrizable ANRs
Let $X$ be a compact metrizable absolute neighborhood retract, abbreviated $ANR$ , meaning a metrizable space that is a neighborhood retract whenever embedded as a closed subset of…
Conjecturesolvedabsolute-neighborhood-retracts geometric-topology homotopy-theoryarXiv posted 2 days ago
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The e-open space conjecture
The e-open space conjecture. Every e-open space is an ANR; equivalently, a space is e-open if and only if it is an ANR.
Conjectureopenabsolute-neighborhood-retracts extension-theory topologyarXiv posted 2 days ago

Borsuk's finite-polyhedron conjecture for compact metrizable ANRs