Let $f:A\rightarrow B$ be a function from a set $A$ to a topology $(B,\tau)$.
Then I can define a subbase on $A$ simply by giving $f[A]\subset B$ the subspace topology inherited from $B$ and defining the inverse image of each open set in $f[A]$ to be open in $A$. Thus let $(A,f^{-1}\tau)$ be the topology on $A$ generated by this subbase.
Question 1: Is this subbase actually a topology? I think probably so but I'd prefer to just have my intuition verified rather than going through the details myself.
Question 2: Is the quotient topology $A/\sim$ given by the relation $x\equiv y$ iff $f(x)=f(y)$ homeomorphic to $B$? I'm almost certain this is true.
Question 3: If we assume $f[A]$ is an open subset of $\mathbb{R}^n$, then $A/\sim$ is a manifold with a single chart and is thus smooth, correct?
Question 4: So assuming question 2 is true, that means that since $[0,1]$ and $\mathbb{R}$ have the same cardinality, I could in theory put a compact topology on the set $\mathbb{R}$, or a non-compact topology on the set $[0,1]$, correct?