# How big can a quotient space be?

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.

• Take $Y$ to be a single point. I'm not sure I follow your example for where the inequality is reversed: both are second countable. Maybe I'm misunderstanding the definition of weight. Jan 7, 2021 at 19:31
• @RobertBell the weight of a space $X$ is the minimal (infinite) cardinality of a base (of open sets) of $X$. I am looking for a quotient map $q:X\rightarrow Y$ such that $w(Y)>w(X)$ by arbitrarily large amounts. Jan 7, 2021 at 19:36
• @RobertBell: I originally had your same objection regarding second countability, because I was interpreting $\mathbb{R}/\mathbb{Z}=S^1$. That is, I was thinking of the Lie theoretic quotient. I belive Tyrone means the topological quotient, so $\mathbb{R}/\mathbb{Z}$ is the countable wedge of circles. Jan 7, 2021 at 19:42
• Yes! $\mathbb{R}/\mathbb{Z}$ is a countable wedge of circles and is not first-countable at the wedge point, so cannot be second-countable. Writing $\mathbb{R}/\mathbb{N}$ gives more or less the same example but may be less/more notationally confusing. Jan 7, 2021 at 19:46

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 $$|B|=|\{q[a]:a\in A\}\le |A|\le 2^{w(X)}.$$
• The weight of $\Bbb R /\Bbb N$ can easily be shown to be equal to a set-theoretic value $b$ called the Bounding Number. If ZFC is consistent then so are (ZFC $\land b<2^{\aleph_0})$ and (ZFC $\land b=2^{\aleph_0}$) Jan 8, 2021 at 1:43