Types of almost continuity

Abstract:

The differences for properties of the range space $f(X)$ are considered for functions $f:X \to Y$, where $f$ is almost continuous relative to $X \times Z$, and $f(X) \subset Z \subset Y$. It is shown that if $f$ is allowed to be almost continuous relative to $X \times Z$, where $X$ is a Peano continuum and $Z$ is a locally connected metric space, then $f(X)$ can be any type of subcontinuum of $Z$. This contrasts the known results for the case where $Z = f(X)$ and almost continuity is relative to $X \times f(X)$. Outer almost continuous retracts ($f:X \to X$ is almost continuous relative to $X \times X$) and inner almost continuous retracts ($f:X \to X$ is almost continuous relative to $X \times f(X))$) are defined. Properties of outer almost continuous retracts, including the existence of an outer almost continuous retract $M$ of a fixed point space $X$, where $M$ does not have the fixed point property, are found.