Example: Examples of Convergent Sequences in Topological Spaces

(related to Chapter: Sequences and Limits)

Example 1

In any space $X$ with the discrete topology $(X,\mathcal P(X))$ only those sequences $(x_n)_{n\in X}$ are convergent to a point $x\in X$ which are constant $x_n=x$ for all indices $n\ge N$ and some first index $N\in\mathbb N.$ This is because the singleton $\{x\}$ is an open set containing $x.$ Therefore $x_n\to x$ if and only if $x_n\in \{x\}$ for all but finitely many $n\in\mathbb N.$

Example 2

In any space $X$ with the indiscrete topology $(X,\{\emptyset,X\})$ all sequences $(x_n)_{n\in X}$ are convergent to every point $x\in X.$ This is becaus $X$ is the only open set containing every point $x$ in this topology. $X$ contains also all sequence members.

Parts: 1

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  1. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
  2. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition