Proposition

Let $X$ be a topological space and let $S$ be a subspace of $X$. Show that the inclusion map $S \rightarrow X$ is a topological embedding.

Solution

Let $i_S$ denote the inclusion map.

• Injective?
  • $i_S(a) = i_S(b) \implies a = b$.
• Continuous?
  • Let $U \subset X$. Then $i_S^{-1}(U) = S \cap U$, and $S \cap U$ is open in the subspace topology $S$.
• Homeomorphism?
  • $i_S$ maps $S$ into $i_S(S) = S$. $i_S$ is the identity map when restricted to $S$. Therefore, it is a homeomorphism between $S$ and $S$.

Therefore, $i_S$ is a topological embedding.