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 $X$.
Best Answer
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 $\prod_{i\in\omega}X_{i}$.