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Proposal : Create tag darboux-function
A function $f:(X,\tau)\to (Y,\tau')$ is a Darboux function if $f(C)\subset Y$ is connected whenever $C\subset X$ is connected.Examples include all continuous functions.
If we restrict our attention to the real valued functions of real variable then functions...
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Q: Is the quotient space of a vector space with a dense subspace isomorphic to the underlying field?
Let $\mathbb{K}$ be a field, $V$ be a $\mathbb{K}$-Banach space and $U<V$ be a dense subspace (with $\overline{U}=V$). I have heard, that:
$$V/U\cong\mathbb{K}.$$
Is this result even correct and if not, what is a simple counterexample? If it is, how do you prove it? Unfortunately, it looks a lot ...
