# Remainder of a space

$X$

The set $Y\setminus X$, where $Y$ is a compactification of $X$. The properties of a remainder depend strongly on those of $X$: A remainder is compact if and only if $X$ is locally compact; the existence of a zero-dimensional remainder depends broadly on whether $X$ has the peripheral compactness property; if there exists a metrizable compactification of $X$ with remainder of dimension $\leq k$, then $X$ has an open base in which the intersection of the boundaries of any $k+1$ disjoint sets is compact; etc. If every connected compact subset of $Y\setminus X$ consists of a single point (for example, if $\operatorname{ind}(Y\setminus X)=0$), the remainder is said to be punctiform. If there exists at least one compactification with punctiform remainder, then one of these compactifications, $\mu X$, is maximal, and, moreover, it is the minimal perfect extension of $X$.

Some important properties, such as compactness, local compactness, paracompactness, the Lindelöf property, are possessed by all remainders of $X$ or by none. When the remainders of $X$ have such a property, $X$ is said to have the property "at infinity" .