TITLE: Inverse limit of an inverse system of topological spaces
QUESTION [9 upvotes]: Given an inverse system $\mathcal G=\{X_i\}$ of topological spaces over some directed set $I$.
If $X=\prod\limits_{i\in I}X_i$, the inverse limit  $X^*=\varprojlim X_i$ of $\mathcal G$ is a subspace of $X$
Could someone explain this to me in a very basic way (I have read many references but could not get it). How $X^*$ is a subspace of $X$.

REPLY [8 votes]: As you tag (category-theory), I will assume you have basic knowledge about it.
The forgetful functor $\mathsf{Top} \to \mathsf{Set}$ admits a left adjoint (namely the discrete topology functor $\mathsf{Set} \to \mathsf{Top}$), so it commutes with (small) limits. Then the underlying set of the limit in $\mathsf{Top}$ is the limit in $\mathsf{Set}$ of the underlying space.
Now, either you know that in $\mathsf{Set}$, every limit $\lim_J F$ is a subset of $\prod_{j \in \mathrm{Ob}\, J} F(j)$, either you show it (once you know that you search such a subset, it is not that hard to guess it ; then show it is indeed the limit).

Edit. Ok, you did not tag (category-theory), so my apologies if it is obscure to you. I leave it nevertheless for those whose land on this page and would be interesting in such an answer.