Question

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$.

Could someone post some left-separated space to help me understand such definition?

Answer 1

This is just filling in some details of Ramiro de la Vega's comment: Given any space $X$, form a (possibly transfinite) sequence of points by the following induction. As long as (the range of) the sequence you've built so far isn't dense in $X$, choose arbitrarily a point not in its closure, and append the chosen point to your sequence. Stop only when your sequence has become dense in $X$. Every initial segment of (the range of) your sequence is closed because, after it was formed, you only added points outside its closure.

Answer 2

The simplest example of a left-separated space would be a countable $T_{1}$-space. If $X$ is a countable $T_{1}$-space, then if $<$ is a well ordering on $X$ such that $(X,<)$ is order isomorphic to the natural numbers, then every initial segment in $X$ is closed. For instance, $\mathbb{Q}$ is left separated. We may generalize this example to get more examples of left-separated spaces with good separation axioms without messing with ordinals.

If $\kappa$ is a regular cardinal, then a $P_{\kappa}$-space is a topological space where the intersection of less than $\kappa$ many open sets is open. $P_{\kappa}$ spaces are easy to come by since one can transform any topological space $X$ into a $P_{\kappa}$-space $(X)_{\kappa}$ by declaring the intersection of less than $\kappa$ many open sets to be open. If $\kappa$ is a regular cardinal and $X$ is a $T_{1}$ $P_{\kappa}$-space of cardinality $\kappa$, then $X$ is left-separated by the same argument as with countable $T_{1}$-spaces. In particular, if $X$ is a $T_{1}$-space (Hausdorff, regular) with $|X|\leq\kappa$, then $(X)_{\kappa}$ is left-separated (and Hausdorff, regular).

Of course, if you do like messing with ordinals, then one can put a left-separated topology on an ordinal $\alpha$ by making sure that every point $\beta<\alpha$ has $[\beta,\alpha)$ as a neighborhood. In particular, if $Z_{\beta}$ is a filter on $\alpha$ with $[\beta,\alpha)\in Z_{\beta}$ for each $\beta<\alpha$, then we may topologize $\alpha$ by letting $U\subseteq\alpha$ be open if $U\in Z_{\beta}$ whenever $\beta\in U$. This topology will be left-separated and any left-separated space can be formed by this construction.