$\varepsilon$-retracts, Q-manifolds, and fixed points
In the class of compact metric spaces, we define the absolute neighborhood $\delta$-retract ($ANR_\varepsilon$-space), a generalization of the corresponding notion from . One of the Noguchi fixed point theorems is generalized with the use of $ANR_\varepsilon$-space, from which the classic Brouwer fixed-point theorem follows. It is proved that there exists a compact noncollapsible acyclic $Q$-manifold with the fixed point property, while it was previously known that a compact non-acyclic $Q$-manifold existed. We prove that if the Cartesian product of a sequence of non-singleton compact metric sets belongs to $ANR$,
then it is a compact $Q$-manifold. A topological space with the fixed point $\sigma$-property is introduced and studied. In particular, we prove that an $ANR$-compact $X$ has the fixed point $\sigma$-property if and only if $X$ is connected. An example of a noncompact set in $R^2$ with the fixed point property is given.
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