Convergence of Sequence in Subspace

Theorem

Let $T = \struct{S, \tau}$ be a topological space.

Let $T_H = \struct{H, \tau_H}$ be a topological subspace of $T$, where $H \subseteq S$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $S$.


Then $\sequence {x_n}_{n \mathop \in \N}$ converges to $p$ in $T_H$ if and only if $\sequence {x_n}_{n \mathop \in \N}$ converges to $p$ in $T$.

Proof


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Sources

  • 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces
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