8 problems

Let $X$ be a closed subset of ${\mathbb R}^n$, let $p\in{\mathbb N}$, and let $f:X\to{\mathbb R}$. The paratangent bundle of order $p$ is denoted by $\tau^p(X)$, and applying the p…

Let $K$ be a countable CW complex, and let $AE(X)$ mean that every map from a closed subset of $X$ to $K$ extends over $X$. A metrizable compactification $\widetilde{X}$ is a compa…

Let $K$ be a countable CW complex. Write $AE(X)$ for the property that every map from a closed subset of $X$ to $K$ extends over $X$. The class of compacta … is said to have a univ…

Let $\omega$ be a definable modulus of continuity, let $X\subseteq\mathbb{R}^n$ be a closed definable set, and let $f$ belong to the trace space…

Graph extension conjecture. The following are equivalent:

Sharp finiteness number conjecture. We conjecture that

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.

Let $G$ be a finite graph with no sinks or sources such that $C^*(G)$ is simple, and let $E_1$ and $E_2$ be one-sink extensions of $G$. Write $[E_i]$ for the class of the extension…