Are point-free topologies a "proper" generalization of pointwise topologies?

Question

I just learned the definition of point-free topologies (also known as pointless topologies). Is every point-free topology expressible as a pointwise topology (so the notion of point-free topologies is nothing but an alternative way to look at existing pointwise topologies), or are there point-free topologies that are not pointwise topologies (so that point-free topologies is indeed a "proper" generalization)?

Answer 1

"Pointfree topologies" (usually called locales) are in some ways more general than "classical" topological spaces, but in other ways aren't. First, let me answer the main point (pun intended) of your question:

There are nontrivial locales that have no points at all. Obviously such a locale cannot have come from a topological space, since the only topological space without points is the empty space.

For example, for any set $X$ you can look at the locale of surjections $\mathbb{N} \to X$. In case $X$ is uncountable, obviously no such surjection exists, so this locale has no points. Yet you can still work with "open sets" of this locale, which tell you properties that hypothetical surjections would have. This is useful because of its close connection to set theoretic forcing -- we can use such a locale to build a new model of set theory (really a topos) in which such a surjection does exist!

However, it's not true that the category of topological spaces sits nicely inside the category of locales. This is basically because a locale only remembers the information about open sets. So two topological spaces with the same open sets will look the same in the localic world, even though they may differ when you look at the points! For example, any space with the indiscrete topology looks the same as a single point in the localic world.

So then, in order to compare locales and topological spaces, both sides need to make some small concessions.

On the localic side, we say a locale is Spatial when it "has enough points" in a sense that can be made precise. The spatial locales, as you might guess, are precisely the locales coming from classical topological spaces.

On the topological side, we say a space is Sober when its points are "determined by its open sets", again in a sense that can be made precise. This condition roughly says that the kind of counterexample we saw with the indiscrete topology doesn't happen, and you won't be surprised to hear these are the spaces which can be recovered from their locale of open sets.

It turns out that the category of spatial locales is equivalent to the category of sober spaces! So if you restrict attention to "nice" spaces, then you get precisely the same objects as the locales "with enough points". In that sense, then, the general locales (possibly without points) form a strict generalization of the sober spaces. In practice most spaces that people care about are sober (for instance, all hausdorff spaces, as well as ring spectra are sober), so it's not too much of a fib to say that "pointless" spaces generalize the classical topological spaces. (But, of course, it is a fib! The nonsober spaces are still important in classical topology, and they have no localic analogue)

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I hope this helps ^_^