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When working with topological vector spaces (say $X,Y$), the term “embedding” is often used for a continuous injection $f:X\rightarrow Y$.

Now, $f$ is of course a bijection onto its image, but it's not necessarily a homeomorphism (as we would normally require of an embedding in topology).

Can anyone explain to me exactly what we're loosing by not having $X$ and $f(X)$ homeomorphic? And is there something special about linear spaces that makes whatever we're missing (more or less) irrelevant?

I suppose that the subspace topology on $f(X)$ must in some sense be coarser than the topology on $X$ if the inverse of $f$ (defined on the range of $f$ ) fails to be continuous

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  • $\begingroup$ in case of topological space if $X$ is compact or $f$ is a proper map or a closed map then continuous injection is an embedding $\endgroup$ – Anubhav Mukherjee Mar 7 '15 at 12:47
  • $\begingroup$ Where does the algebraic structure becomes involved here? (It seems the question is about merely topological spaces.) $\endgroup$ – C-Star-W-Star Mar 8 '15 at 20:08
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Think of the torus as the quotient of $\mathbb R^2$ in the usual way: $(x, y) \mapsto (x \bmod 1, y \bmod 1)$.

Now look at the line $y = \pi x$ in the plane. Under this quotient, it becomes a curve in the torus, and the map is 1-1 and continuous, but locally, in the image, it doesn't look the way you expect, i.e., like a line in $\mathbb R^2$, where near the line there is "no other stuff".

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As $X$ and $f(X)$ are not homeomorphic, if we identify $X$ with $f(X)$, the topology on $X$ is not the same as the subspace topology, so that one has to deal with the topology on $X$ seperately. This is important e.g. in the difference between immersed and embedded submanifolds.

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Both refer to induced structures...

Subspace* and embedding: $$\iota:S\hookrightarrow\Omega:\quad\mathcal{T}(S)=\{U\subseteq S:U\in\iota^{-1}\mathcal{T}(\Omega)\}$$ (The embedding being a homeomorphism onto its image.)

Projection and quotient space: $$\pi:\Omega\twoheadrightarrow\tilde\Omega:\quad\pi^{-1}\mathcal{T}(\tilde\Omega)=\{\tilde{U}\subseteq\tilde\Omega:\pi^{-1}\tilde{U}\in\mathcal{T}(\Omega)\}$$ (The projection being an open continuous map.)

...There are many more considerable structures.

*Subspace in the general sense: $S\nsubseteq\Omega$

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