Does the dimension of a manifold depend on the topology? That is, can I endow a set with a topology $T$ and get an $n$-dimensional manifold, and endow the exact same set with another toplogy $T'$ and get an $m$-dimensional manifold, with $n\neq m$?
Can a curve like this:
$\hskip1.3in$
be given the topology of a 1-manifold? Or a curve like this cannot be a manifold with any topology?