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Consider a topological space $(X,\tau)$. I would like to know whether the set $\tau$ of open sets can be given its own topology in a canonical way. In that respect I'm asking for the same kind of thing as this previous question. However I'd also like to specify a property that this canonical topology should have, which the other question doesn't do, so I hope this won't be considered a duplicate.

Specifically, let $(X,\tau)$ and $(Y,\sigma)$ be topological spaces, and consider a continuous function $f\colon X\to Y$. We can define the pullback map as a function $f^*\colon \sigma\to\tau$ that maps open subsets of $Y$ to their preimages under $f$, which are open subsets of $X$.

I would like to know whether $f^*$ can itself be seen as a continuous function. That would mean defining the open subsets of $\sigma$ and $\tau$. I'm wondering whether these can be defined in a canonical way, such that every continuous map between any pair of topological spaces has a continuous pullback.

Of course this would be the case if we just defined the topology on $\tau$ to be the discrete topology for every topological space $(X,\tau)$, but I'm wondering if it's possible to do it in a way where they can be coarser than that.

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    $\begingroup$ Open subsets can be identified with maps to Sierpinski space and you can consider the compact-open topology on this (en.m.wikipedia.org/wiki/Compact-open_topology). I don’t know anything about it though. $\endgroup$ – Qiaochu Yuan Jan 21 at 6:42
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    $\begingroup$ Fun puzzle! If $X$ is a nice space, then one might hope that a small open neighbourhood of $U$ in the canonical topology will consist of opens that are relatively 'close' to $U$ in the intuitive sense. $\endgroup$ – Jeroen van der Meer Jan 21 at 10:38
  • $\begingroup$ An easy way to give $\tau$ a topology is to let $2$ be the discrete space with two points and realise it as a subspace of $2^X=\prod_{x\in X}2$. This maps the maps $f^*$ contintuous and produces many interesting examples. $\endgroup$ – Tyrone Jan 21 at 14:17
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There are many ways to topologize the set $\tau$. You can give it the trivial topology or the discrete toplogy, but surely that is not what you expect.

Another way to do it is to observe that $\tau$ is partially ordered by set inclusion. With respect to this partial order it is a lattice. Now have a look at Compact subspaces of the Poset.

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