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What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open sets such that for open $U,V \in X$:

$\ Hom_{Op_X}(U,V) = \begin{cases} \{pt\} & U \subset V \\ \emptyset & \text{otherwise} \end{cases} \ $

Thus for the category of open sets $Op_X$ on $X$, the objects are open sets (i.e. $U,V$) and the morphisms are inclusions $U \subset V$.

Now however if we have say $U \subset V \subset W$ then $Hom_{Op_X}(U,V) = \{pt\}$ but $Hom_{Op_X}(V,W) = \{pt\}$. What does this mean i.e. what is the point (no pun intended)? Furthermore can we seperate these points?

What I am trying to get at (although I am really wanting to understand the general case above and the $\{pt\}'s$ But what if I had a set say $\{1,2,3,4,5,6,7\} = W$ another $\{2,3,4,5\} = V$ and yet another $\{2,3,4\} = U$ so as to satisfy the inclusion $U \subset V \subset W$, then what is going on if both of the inclusions $U \subset V$ and $V \subset W$ are equal to $\{pt\}$?

Thanks,

Brian

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    $\begingroup$ You are overcomplicating this, I think. The category you want is simply the subcategory of the category of topological spaces whose objects are the open subsets of $X$ and whose morphisms are the inclusion maps. $\endgroup$ Nov 22, 2013 at 4:01
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    $\begingroup$ Since in this subcategory between two objects there is at mnost one morphism, whatever source you are getting this from chooses not to name the morphisms and uses the name 'pt' for all of them. This does not intoduce an ambiguity. $\endgroup$ Nov 22, 2013 at 4:02

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The open sets of a topological space form a lattice, which is a certain type of poset. We can turn posets into a fairly simple type of category.

A poset, or partially ordered set is a set with a binary relation $(S, \le)$ that satisfies:

Reflexivity: $a \le a$

Symmetry: if $a \le b$ and $b \le a$, then $a = b$

and Transitivity: if $a \le b$ and $b \le c$, then $a \le c$.

The open sets of a topological space with inclusion as the relation $(\tau, \subset)$ satisfy the axioms of a poset (you can verify that yourself).

To turn a poset $(S, \le)$ into a category $\mathcal{S}$, take $\mathrm{ob}(\mathcal{S}) = S$, and for any two objects $a$ and $b$, there exists a unique arrow between them $a \rightarrow b$ if and only if $a \le b$. That is, $\hom(a,b) = \{\rightarrow\}$ if $a \le b$ and $\hom(a,b) = \emptyset$ otherwise. This is a category because we have the identity (reflexivity), and composition of arrows (transitivity), and any two arrows $a \rightarrow b$ and $b \rightarrow a$ must be isomorphisms (symmetry - this is to preserve the uniqueness of arrows from an object to itself). These categories essentially look like Hasse Diagrams.

I think you are getting confused over the contents of the hom sets. They just contain a 'pt' if there is an arrow between the two open sets in question. This is not a point, it is a unique arrow. I would call the element 'ar' or '1' or '$\rightarrow$' or something.

Posets as categories are really cool, because they are really simple but categorical constructions in them are meaningful. For example, products and coproducts are meets and joins. Adjoint functors between poset categories are Galois connections. If a poset category has binary (non-empty) products and coproducts, then it is a lattice. If it has a terminal and initial object, then it is a bounded lattice. If it is bicomplete, then it is a complete lattice, etc...

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