Inverse limit of an inverse system of topological spaces

Question

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 $X$.

Answer 1

The product $\prod_{i\in\omega}X_{i}$ is the set of all sequences $(x_{0},x_{1},...)$ where $x_{i}\in X_{i}$ for each $i$ .

Now you are given a family of continuous “bonding” maps $f_{i}:X_{i+1}\to X_{i}$ .

The inverse limit is a special subset of $\prod_{i\in\omega}X_{i}$ , consisting of those sequences $(x_{0},x_{1},...)$ satisfying $f_{i}(x_{i+1})=x_{i}$ for each $i$ . I like to think of the elements of the inverse limit as “threads” because the terms are linked together by the functions $f_{i}$ . If you picture this from $X_{0}$ going up, you get a tree-like structure.

Now give this subset the subspace topology, where $\prod_{i\in\omega}X_{i}$ has the product topology. That is, $U$ is open in $X^*$ iff $U=V\cap X^*$ for some open $V$ in $\prod_{i\in\omega}X_{i}$.

Answer 2

You just construct $\varprojlim_i X_i$ as a subspace of the product $\prod_i X_i$, namely consisting of those tuples which are compatible with respect to the transition maps. This even works for every index category, not just for directed partial orders.

Answer 3

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.