How big can a quotient space be?

general-topology

Solution 1:

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 $|B|=|\{q[a]:a\in A\}\le |A|\le 2^{w(X)}.$

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