Determining Left Adjoint of Forgetful Functor from $\tau_{*}$ to $\tau$

Question

This is a homework problem so I ask only for a small push in the right direction.

I am asked to determine the left adjoint of the forgetful functor $F: \tau_{*} \to \tau$ of pointed topological spaces to topological spaces.

For any pointed topological space, $(X, x_0)$, I defined $F(X, x_0) = X$ and, for any $f \in Hom((X, x_0), (Y, y_0))$, I defined $F(f) = f \in Hom(X, Y)$.

The way I began thinking about this is, every continuous function, $f$ between two topological spaces $X, Y$ is also a continuous function between two pointed topological spaces $(X, x_0), (Y, f(x_0))$. However, there are multiple continuous functions between $X, Y$ that are also continuous functions between $(X, x_0), (Y, f(x_0))$. Am I looking at this set up incorrectly?

Answer 1

$G:\tau_*\leftarrow \tau$, the functor defined by taking topological space $X$ to the disjoint union $X\cup {•}$ of $X$ with a one point set, which will be taken as the base point... is the left adjoint functor to the forgetful functor, in this case... See https://en.m.wikipedia.org/wiki/Pointed_space ...

Apparently, this assignment satisfies a universal property...

Answer 2

The intuition I use for determining how free functors should behave is that they take whatever object you give them and add in just as much junk as they need to equip that object with the desired structure. For example, consider the 'free group' functor, which is left adjoint to the forgetful functor $U : \mathbf{Grp} \to \mathbf{Set}$. The free group on a set $X$ is defined by taking all the elements of $X$ and doing everything 'groupy' you can do to them; namely, its elements are all the formal products (concatenations) of elements of $X$ and their (formal) inverses, and so on.

In this case, given a topological space $X$, you need to add in as much junk into $X$ as is required in order to turn it into a pointed topological space. Since a pointed topological space is just a topological space with a distinguished choice of point, it is natural to suspect that the left adjoint to the forgetful functor $\mathbf{Top}_{\star} \to \mathbf{Top}$ assigns to each space $X$ the topological space whose underlying set is $X \cup \{ \star \}$ with distinguished point $\star$, where $\star \not \in X$, and whose open sets are the open sets in $X$ and $\{ \star \}$.

You need to extend this definition to continuous maps and prove that it really does define a left adjoint to the forgetful functor.