\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
deepdream of
          a sidewalk
Show Question
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)
Math and science::Topology

Sequence convergence

Sequence convergence

Let \( X \) be a topological space, let \( (x_n) \) be a sequence in \( X \) and let \( x_0 \in X \). Then \( (x_n) \) converges to \( x_0 \) iff for every neighbourhood \( U \) of \( x_0 \) there is an \( N \ge 1 \) such that for every \( n \ge N \), \( x_n \in U \).

In other words, every neighbourhood of the point to which a sequence converges must contain the whole tail end of the sequences starting from some arbitrary point in the sequence.

It is a good exercise to check that this definition carries the expected meaning for metric spaces.


Sequences in Hausdorff spaces

A common demonstration of the importance of Hausdorff spaces:

Let \( X \) be a Hausdorff topological space. Then each sequence in \( X \) converges to at most one point.

Another good proof exercise.

Multiple convergence

An example where we get multiple convergence: