Exercise 2.2.3 (Convergence in relative topology)

Question

Let $S$ be a subset of a topological space $X$. Show that a sequence $\left\{x_{i}\right\}$ in $S$ converges to $x_{0} \in S$ in the relative topology if and only if, considered as a sequence in $X$, the sequence $\left\{x_{i}\right\}$ converges to $x_{0}$.

Elu Thingol · Accepted Answer

This follows directly from the definition of relative topology. In other words,
\begin{equation*}
\begin{array}{lll}
&\{x_i\} 	ext{ converges to $x_0$ in $\mathcal{T}_Xestriction_S$}\
& \iff 	ext{ for all open sets } U \in \mathcal{T}_Xestriction_S 	ext{ containing } x_0  &\exists N \in \mathbb{N} \quad \forall i \geq N: \quad x_i \in U\
& \iff 	ext{ for all open sets } U \in \mathcal{T}_X 	ext{ containing } x_0 &\exists N \in \mathbb{N} \quad \forall i \geq N: \quad x_i \in U \cap S\
& \iff 	ext{ for all open sets } U \in \mathcal{T}_X 	ext{ containing } x_0 &\exists N \in \mathbb{N} \quad \forall i \geq N: \quad x_i \in U\
&\{x_i\} 	ext{ converges to $x_0$ in $\mathcal{T}_X$},&
\end{array}
\end{equation*}
where in the last step we have used the fact that $\{x_i\}$ is contained in $S$.

Exercise 2.2.3 (Convergence in relative topology)

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Exercise 2.2.3 (Convergence in relative topology)

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