Topological Embedding

Last updated Nov 1, 2022

Let $X, Y$ be Topological Spaces and let $f: X \to Y$. $f$ is a Topological Embedding into $Y$ if $f: X \to f(X)$ is a Homeomorphism, where $f(X)$ is given the Subspace Topology in $Y$.

1. A Topological Embedding is necessarily continuous. Indeed if $U \subset Y$ is Open, then $f^{-1}(U) = f^{-1}(U \cap f(X))$, which is Open because $U \cap f(X)$ is Open in the Subspace Topology and $f: X \to f(X)$ is a Continuous Function by definition of a Homeomorphism.
2. A Topological Embedding is necessarily injective. Indeed if $x_{1}, x_{2} \in X$ so that $f(x_{1}) = f(x_{2})$, then because $f(x_{1}), f(x_{2}) \in f(X)$ and $f$ is a Bijection onto $f(X)$ by definition of a Homeomorphism, $x_{1}= x_{2}$.