Definition. Let $X$ and $Y$ be topological spaces. Suppose $f:X\to Y$ is an injective continuous map. If the function $f':X\to\ f(X)$ obtained by restricting the range of $f$ is a homeomorphism, then the map $f:X\to Y$ is called a topological embedding.
In defining topological embeddings, is it necessary to first suppose $f:X\to Y$ is an injective and continuous? Is there such a function $f:X\to Y$ that is not injective or continuous, but the function $f':X\to f(X)$ is a homeomorphism?
The requirement $f:X\to Y$ be injective and continuous seems redundant to me, for if the function $f':X\to\ f(X)$ obtained by restricting the range of $f$ is a homeomorphism, then it automatically guarantees $f$ is continuous and injective. In short, why not define a topological embedding as follows?
Definition. A map $f:X\to Y$ between topological spaces is called a topological embedding if the function $f':X\to\ f(X)$ obtained by restricting the range of $f$ is a homeomorphism.
Best Answer
The last definition I think captures the essence of being an embedding. It indeed implies the original $f$ is continuous (provided we assume $f[X]$ has the subspace topology wrt $Y$, as we must) and injective.
Definitions are not always supposed to be as tight as possible. Here the author wanted to frame the discussion, as it were, so that the reader would only consider injective continuous maps as candidates. Embeddings are a special subclass of these maps. The discussion could hereafter continue with some examples and non-examples of such embeddings, e.g.