Injectivity of topological embeddings

Lee states that an “injective continuous map that is a homeomorphism with its image is an embedding.”

Is the word “injective” important here? If $f: X \to Y$ is a homeomorphism when we restrict the codomain to $f(X)$, then $f$ must be injective, since homeomorphisms are bijective. So $f$ must be injective by virtue of it being a homeomorphism to its image.

I ask if it’s important because I may have misunderstood some definition (I’m fairly new to topology) and I’m worried I got something wrong.

Your reasoning is correct. The word "injective" wasn't necessary, but might have been used for emphasis.

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