Is every set a pointed set?

Question

My question is quite simple, It seems every non-empty set is a pointed set, only we have to do is choice some element to be the distinguished element, am I right? I'm looking for non-empty sets which aren't pointed sets.

EDIT

I think I didn't express myself clear, my question is can every set be transformed into a pointed set, just picking up a point to be the distinguished point?

Thanks in advance

Answer 1

A non-empty set is not a pointed set until you actually pick a point in it. The pointed set is the combination of a set and a particular element in it.

So $\{1,2\}$ is not a pointed set, but $\langle \{1,2\},1\rangle$ and $\langle\{1,2\},2\rangle$ are two different pointed sets that can be made from it.

And if somebody talks about "pointed sets" rather than just "nonempty sets", it must be because exactly that difference is important to them in the context!

Answer 2

There is a functor from the category of pointed sets to the category of sets, that is, forgetting the chosen point. However there's no “canonical” functor backwards unless you define it by choosing anything which is not a set and add it.

Actually the category of pointed sets is equivalent to the category of sets with partial maps, which is not equivalent to the category of sets with (everywhere defined) maps.

What's a partial map $f$ from $A$ to $B$? It's a quadruple $(A,A',f,B)$ where $A'\subseteq A$ and $f$ is a (usual) map $f\colon A'\to B$.

The equivalence is readily established: map every pointed set $(A,a)$ to $F(A)=A\setminus\{a\}$ and a map of pointed set $f\colon (A,a)\to (B,b)$ to the partial map $(F(A),A'=A\setminus f^{-1}(\{b\}),f|_{A'},F(B))$.

This is readily seen to be a functor from $\mathbf{PSet}\to\mathbf{Set}$. The “inverse” functor is obtained by defining $G(A)=A\cup\{*\}$, where $*$ is a formal symbol equal only to itself and not belonging to any set; if $p=(A,A',f,B)$ is a partial map from $A$ to $B$, define $G(p)$ to be the map $G(p)\colon G(A)\to G(B)$ such that $$ G(p)\colon x\mapsto \begin{cases} f(x)&\text{if $x\in A'$},\\ * & \text{if $x\in A\setminus A'$},\\ * & \text{if $x={*}$}. \end{cases} $$

Verifying that $G$ is a functor and that $(F,G)$ is an equivalence of categories is easy.

So, no, a non empty set is not a pointed set. You can choose a point so that it becomes a set, but nothing guarantees that when you have a map $f\colon A\to B$ of non empty sets, the choices you do in $A$ and $B$ makes this map into a map of pointed sets.

Answer 3

Every single nonempty set can be made into a pointed set by choosing out an arbitrary element. A collection of infinite number of nonempty set, however, need not be capable of all being made into pointed set, if Axiom of Choice is not accepted.

Consider an index set $I$ that is infinite. Let $A_{i}$ be a collection of nonempty set, and let $B_{i}$ be a collection of pointed set. Assume an universe where Axiom of Choice is not true. Then:

$\prod_{i\in I}A_{i}$ can be an empty set. But $\prod_{i\in I}B_{i}$, as a set, is nonempty; in fact it contains at least one point, with each coordinate being the distinguished point in each point set.

So there are huge differences. Choosing can be very hard indeed.