Definition 4.2.13 (Limit). Let $X, Y$ be topological spaces, $A \subset X$, and $f: A \to Y$ be a function. For any filter base $\fB \subset 2^{A}$, if $f(\fB)$ converges to $y \in Y$, then $y = \lim_{x, \fB}f(x)$ is a limit of $f$ with respect to $\fB$.

For any $x_{0} \in \overline{A}$, let $\fF = \bracs{U \cap A| U \in \cn(x_0)}$ be the trace of $\cn(x_{0})$ on $A$, then $\fF \subset 2^{A}$ is a filter. If $f(\fF)$ converges to $y \in Y$, then

is a limit of $f$ at $y$ with respect to $A$. If $A = X$, then $x \in A$ may be omitted.