Let $(I,\preceq)$ be a directed poset, i.e. $I$ is a set and $\preceq$ is a binary relation on $I$ with the following conditions:
- $i \preceq i$ for all $i \in I$,
- $i \preceq j$ and $j \preceq k$ imply $i \preceq k$ for all $i,j,k \in I$,
- $i \preceq j$ and $j \preceq i$ imply $i=j$ for all $i,j \in I$,
- for any $i,j \in I$, there exists $k \in I$ with $i,j\preceq k$.
Definition: Let $(I,\preceq)$ be a directed poset. An inverse system of topological spaces on $I$ is a collection $\{X_i \,|\, i \in I\}$ of topological spaces and $\{ \varphi_{i,j}: X_i \to X_j \, | \, i \succeq j \}$ of continuous maps such that $\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}$ for every $i \succeq j \succeq k$.
In order to understand this definition, I wanted to construct an example which is not the constant inverse system. The example I came up is the following:
The directed poset shall be $(\mathbb{N},\leq)$, the sets $X_n = \{1,\dots,n\}$ shall be endowed with the discrete topology and $\varphi_{mn}:X_m \to X_n$ shall be defined by $\varphi_{mn}(x) = \min\{x,n\}$ for any $m \geq n$. I think that my example is an inverse system on $\mathbb{N}$.
Is there a way to simplify or generalize this example?
Thank you in advance!
Best Answer
If $\mathcal{A}$ is a family of subsets of a space $X$ (in the subspace topology) which is closed under finite intersections, we can see it as a directed poset: $A_1 \le A_2$ iff $A_2 \subseteq A_1$. The intersection of $A_1$ and $A_2$ is their common upperbound as required by the 4th property.
If $A_1 \le A_2$ we have a natural inclusion map $i_{A_2,A_1}: A_2 \to A_1$ defined by $i_{A_2,A_1}(x)=x$ which is of course continuous (in the respective subspace topologies). This obeys all the axioms for an inverse system.
As an exercise you can check that the limit for this system is just $\bigcap \mathcal{A}$. (which need not be non-empty of course, but the empty set is a valid, somewhat boring, space; it will be non-empty in the case of non-empty closed subsets of a compact Hausdorff space, e.g.)
Another instructive example: let $I$ be some index set and for each $i \in I$ we have a space $X_i$. Let the poset $P$ be all finite non-empty subsets of $I$ ordered by inclusion. This is clearly a directed poset.
For each $p \in P$ we construct the finite product space $X_p:=\prod_{i \in p} X_i$ and if $p \le p'$ we have a natural continuous projection map $\pi_{p',p}: X_{p'} \to X_p$ (which is a function restriction really). This again defines an inverse system whose limit is homeomorphic to $\prod_{i \in I} X_i$.
Both examples give an idea of the fact that the spaces in the inverse system are a sort of approximation of the limit space, and the approximation gets better for larger indices of the directed set...