TITLE: How big can a quotient space be?
QUESTION [11 upvotes]: Let $X$ be a space of weight $w(X)=\kappa$. Suppose $q:X\rightarrow Y$ is a quotient map. If $q$ is open, or if $Y$ is compact, then $w(Y)\leq w(X)$. In general it is possible for $w(Y)$ to be larger than $w(X)$ (consider $\mathbb{R}/\mathbb{Z}$).

How much larger than $w(X)$ can $w(Y)$ be?

Really what I'm looking for is a machine to produce countexamples for any fixed infinite cardinal $\kappa$. I'm particularly interested in the case that $X$ is second-countable.

REPLY [3 votes]: Let $T_X$ be the topology on $X.$ Then $|T_X|\le 2^{w(X)}.$
Now $q:X\to Y$ is a continuous surjection. So if $B$ is a base for $Y$ then $A=^{def}\{q^{-1}b: b\in B\}\subset T_X. $
So $|A|\le 2^{w(X)}.$
So $|B|=|\{q[a]:a\in A\}\le |A|\le 2^{w(X)}.$