[Munkres, Ch40, p252] Find a nondiscrete space that has a countably locally finite basis but does not have a countable basis.
I solved this problem. My solution is $X=\mathbb{R} \cup \{0'\}$ with topology given by basis $\mathfrak{B}$ consisting of all the nonzero points and {0,0'}. This is almost discrete, except for zero points. This basis $\mathfrak{B}$ is locally finite, so countably locally finite. And this space has no countable basis, since all the nonzero points(which are uncountable) should be in the basis.
But I think that this space can be regarded discrete by passing to the quotient space. Is there any more truly nondiscrete example?
Best Answer
Note that Munkres’ terminology is non-standard: what he calls countably locally finite is usually called $\sigma$-locally finite.
Here’s an example in which discreteness is used in a more subtle way.
Let $D$ be an uncountable discrete space, and let $C=\{0,1\}^\omega$ be the product of countably infinitely many copies of the discrete two-point space. $C$ is homeomorphic to the familiar middle-thirds Cantor set and is a compact metric space without isolated points. It also has a $\sigma$-discrete (and hence $\sigma$-locally finite) base of clopen sets. Now let $X=D\times C$; then $X$ is easily seen to have a $\sigma$-discrete base as well, but it clearly has no countable base.
Indeed, any space with a $\sigma$-locally finite base could be substituted for $C$. Let $Y$ be such a space, and let $\mathscr{B}=\bigcup_{n\in\omega}\mathscr{B}_n$ be a base for $Y$ such that each $\mathscr{B}_n$ is locally finite. For $n\in\omega$ let $$\mathscr{B}_n'=\Big\{\{x\}\times B:x\in D\text{ and }B\in\mathscr{B}_n\Big\}\;,$$ and let $\mathscr{B}\,'=\bigcup_{n\in\omega}\mathscr{B}_n'$; then $\mathscr{B}\,'$ is a $\sigma$-locally finite base for $X$.
More generally yet, you know from the Nagata-Smirnov theorem that a $T_3$ space is metrizable iff it has a $\sigma$-locally finite base. Thus, every non-second-countable metric space is an example. For example, let $\kappa$ be any uncountable cardinal number, and let $D$ be a set of cardinality $\kappa$. The metric $$d:D\times D\to\Bbb R:\langle x,y\rangle\mapsto\begin{cases}0,&\text{if }x=y\\1,&\text{if }x\ne y\end{cases}$$ generates the discrete topology on $D$. Now let $X=D^\omega$; the function $$\rho:X\times X\to\Bbb R:\Big\langle\langle x_n:n\in\omega\rangle,\langle y_n:n\in\omega\rangle\Big\rangle\mapsto\sum_{n\in\omega}\frac{d(x_n,y_n)}{2^n}$$ is a metric on $X$, and the weight (= minimum cardinality of a base) of $X$ is $\kappa$.