something related to Alexandrov one-point compactification

The topology for $X\bigcup \{\mathrm{\infty }\}$ is defined as follows: there are two kinds of open sets of $X\bigcup \{\mathrm{\infty }\}$. If $\mathrm{\infty }\notin U$, then $U$ is open if and only if $U$ is an open set in the topology of $X$. If $\mathrm{\infty }\in U$, then $U$ is open if and only if $U$ is the complement of a closed compact subset $K$ of $X$.