Editing Net (mathematics) (section)

=== Limits of nets ===

{{anchor|Limit of a net|Limit point of a net|Convergent net|Net convergence}}
A net <math>x_\bull = \left(x_a\right)_{a \in A}</math> is said to be {{em|eventually}} or {{em|residually}} {{em|in}} a set <math>S</math> if there exists some <math>a \in A</math> such that for every <math>b \in A</math> with <math>b \geq a,</math> the point <math>x_b \in S.</math> A point <math>x \in X</math> is called a {{em|{{visible anchor|limit point}}}} or {{em|{{visible anchor|limit|Limit of a net}}}} of the net <math>x_\bull</math> in <math>X</math> whenever:
:for every open [[Topological neighborhood|neighborhood]] <math>U</math> of <math>x,</math> the net <math>x_\bull</math> is eventually in <math>U</math>,
expressed equivalently as: the net {{em|{{visible anchor|converges|Convergent net}} to/towards <math>x</math>}} or {{em|has <math>x</math> as a limit}}; and variously denoted as:<math display="block">\begin{alignat}{4}
                  & x_\bull && \to\; && x && \;\;\text{ in } X \\
                  & x_a     && \to\; && x && \;\;\text{ in } X \\
\lim        \; & x_\bull && \to\; && x && \;\;\text{ in } X \\
\lim_{a \in A} \; & x_a     && \to\; && x && \;\;\text{ in } X \\
\lim_a   \; & x_a     && \to\; && x && \;\;\text{ in } X.
\end{alignat}</math>If <math>X</math> is clear from context, it may be omitted from the notation.

If <math>\lim x_\bull \to x</math> and this limit is unique (i.e. <math>\lim x_\bull \to y</math> only for <math>x = y</math>) then one writes:<math display=block>\lim x_\bull = x \;~~ \text{ or } ~~\; \lim x_a = x \;~~ \text{ or } ~~\; \lim_{a \in A} x_a = x</math>using the equal sign in place of the arrow <math>\to.</math>{{sfn|Kelley|1975|pp=65–72}} In a [[Hausdorff space]], every net has at most one limit, and the limit of a convergent net is always unique.{{sfn|Kelley|1975|pp=65–72}}
Some authors do not distinguish between the notations <math>\lim x_\bull = x</math> and <math>\lim x_\bull \to x</math>, but this can lead to ambiguities if the ambient space ''<math>X</math>'' is not Hausdorff.