Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let $\tau_X\subset\mathcal{P}(X)$ be the topology on $X$.
What are some useful topologies on either $\mathcal{P}(X)$, $\mathcal{P}_0(X)$ or $\tau_X$?
Since $\mathcal{P}(X)$ and $\tau_X$ are complete lattices, I would be interested to know which of the proposed topologies turn them into topological lattices.
Here's what I have so far.
- $\mathcal{P}(X)$ may be topologized as the product space $$\prod_X 2$$ where $2=\lbrace 0,1\rbrace$ with the discrete topology. $\mathcal{P}_0(X)$ may then be given the subspace topology. however, this doesn't use the topology of $X$. The same problem arises when one endows $\mathcal{P}(X)$ with any of the order topologies associated to the partial order $\subset$.
- $\tau(X)\simeq\mathrm{map}(X,\tilde{2})$ may be given the compact open topology, where $\tilde{2}$ is the topological space with underlying set $\lbrace 0,1\rbrace$ and only non trivial open set $\lbrace 1\rbrace$, and the correspondance is given by $O\mapsto\mathbf{1}_O$, the "indicator function" of the open set $O$. In this case it seems, at first glance, that a subbasis is given by the $[K]=\lbrace O\in\tau_X\mid K\subset O\rbrace$ (where $K\subset X$ is any compact subset of $X$). It then holds that $[K]\cap[L]=[K\cup L]$
- $\mathcal{P}_0(X)$ may be given two topologies $\tau_o$ and $\tau_f$ (see the exercises in Bourbaki's first chapter of General Topology). $\tau_o$ is the coarsest topolgy such that for any open subset $O\subset X$, $\mathcal{P}_0(O)\subset\mathcal{P}_0(X)$ is open, and $\tau_f$ is the coarsest topolgy such that for any closed subset $F\subset X$, $\mathcal{P}_0(F)\subset\mathcal{P}_0(X)$ is closed.
My endgoal isn't so much to topologize $\mathcal{P}(X)$ as it is to meaningfully topologize (special subsets of) the set $\mathcal{E}(X)\subset\mathcal{P}(X\times X)$ of all equivalence relations on $X$.
Are there some useful topologies on $\mathcal{E}(X)$ (or special subsets of it)?
There must be some resources in the literature, I'd be happy about some references on the subject.
EDIT 1 Following @ArthurFisher's answer to the question @AsafKaragila links to, one can consider several topologies associated to that of $X$ and the poset structure of $\mathcal{P}(X)$. For any subsets $A\subset B$ of $X$, there are four closed, open and half open intervals $[A,B],(A,B)$ and $[A,B),(A,B]$. Instead of considering the topologies generated by either of the families of
- all open intervals $(A,B)$,
- all half open intervals of the form $(A,B]$
- all half open intervals of the form $[A,B)$
- all half open intervals of the form $(A,X]$
- all half open intervals of the form $[\emptyset,A)$
all of which ignore the topology on $X$, one may restrict oneself to those intervals that are bounded by open, closed or compact sets.