Definition
A subset N of a space X is a neighborhood of infinity if \( \bar{X /N } \) is compact.
We say that X has k ends ( \( k \in \mathbb{N} \cup \{ \infty \} \) ) if \( k = sup \{ j | X \textrm{has a nbd of} \infty \textrm{with j unbounded components} \} \)
Example
- X has zero ends iff X is compact (for nice spaces)
- \( \mathbb{R} \) has at least two ends. (Ex. exactly two)
- \( \mathbb{R}^n, n \geq 2 \) has at least one end. (Ex. exactly one)
- 3 ends
- Infinitely many ends
- Infinitely many ends
Remark: The above does not work well with bad spaces.
Goal: Develop a better theory of ends which will allow us to distinguish the last two examples.
Assume from now on that X is connected, locally connected, locally compact and Hausdorff.
Lemma A
Let \( C \subseteq X \), compact. Then X\C has finitely many unbounded components.
Proof: Let \( \{ U_{\alpha} \}_{\alpha \in A} \) be the set of unbounded components of X\C.
By local connectedness each \( U_{\alpha} \) is open in X \ C, and in X.
Also each \( U_{\alpha} \) is closed in X\C.
By connectedness each \( U_{\alpha} \) has limit points in C.
By local compactness there exists a compact set \( D \subseteq X \) such that \( C \subseteq int D \)
No \( U_{\alpha}\) is contained in D, otherwise it would be bounded.
Claim: No \( U_{\alpha} \) is contained in X \ D.
Otherwise \( \bar {U_{\alpha}} \subseteq \) X \int D which implies \( U_ {\alpha } \) has no limit point in C.
Since \( U_{\alpha} \) is connected and contains points of both D and X\D then \( U_{\alpha} \cup Fr D \neq \phi \)
For each \( \alpha \in A \), choose \( x_{\alpha} \in U_{\alpha} \cap Fr D \).
By compactness, \( \{y_{\alpha} \} \) has a limit point x in Fr D. Then \( x \in \) X\C. Choose a connected nbd V of xlying in X \C
Choose \( x_{\alpha_1 } \neq x_{\alpha_2} \in V \) (uses Hausdorff). Then \( U_{\alpha_1} \cup V \cup U_{\alpha_2 } \) is connected and lies in X \ C.
Remark
It is possible for X \ C to have infinitely many bounded components.
Lemma B
If \( C \subseteq X \) compact, and \( C_0 \) is the union of C with all of the bounded components of X \ C, then \( C_0 \) is compact.
(and its complementary components are precisely the unbounded components of C).
Proof: Choose compact set D, such that \( C \subseteq int D \).
Let \( \mathcal{E} \) be the set of bounded components of X \ C.
& C’ = \( C \cup ( \cup \{ E \in \mathcal{E} | E \subseteq D \} \)
Then \( C’ \subseteq D \) and X \ C’ is open so C’ is closed and hence compact.
claim: All but finitely many elements of \( \mathcal {E} \) lie in D.
Proof: Each element of \( \mathcal{E} \) has its frontier in C. so if \( E \in \mathcal{E} \) does not lie in D it contains points of Fr D.
Now use an argument like before to finish the claim.
Now \( C_0 = C’ \cup \{\cup_{i=1}^n E_i \} \) whose each \(E_i \) is bounded and has frontier in C.
So \( C_0 \) is compact.
Recall: A partially ordered set (poset) is a set A together with a relation \( \leq \) satisfying:
- \( a\leq a \forall a \in A \)
- if \( a \leq b \) and \(b \leq a \) then a = b
- if \( a \leq b \) and \( b \leq c \) then \( a \leq c \)
Ex. For any set X, \( (P(X), \subseteq ) \) is a poset.
Ex. Every subset of a poset is a poset.
Hence if \( \mathcal{W} \subseteq {W} \) then \( \mathcal{W} \) is a poset under \( \subseteq \) relation.
A poset is a pair, \( (A, \leq ) \), is directed, if for any pair of elements if \( \forall a, b, \in A \) there exist \(c \in A \) such that \( a \leq c \) and \( b \leq c \)
Let \( \mathcal{C} \) be a category, for example: Sets, Topological Spaces, Simplicial maps, CW complexes, Groups, Rings, Modules
Inverse System
An inverse system \( \{X_{\alpha} , f_{\alpha}^{\beta}, A \} \) in \( \mathcal{C} \) is
- a directed set \( (A, \leq) \)
- An object \( X_{\alpha} \in \mathcal{C} \) for each \( \alpha \in A\)
- For any \( \alpha \leq \beta \) in A, a morphism \(f_{\alpha}^{\beta} : X_{\beta} \to X_{\alpha} \) satisfying
- \( f_{\alpha}^{\alpha} = id_{X_{\alpha}} \forall \alpha \in A \)
- if \( \alpha \leq \beta \leq \gamma \) then \( f_{\alpha}^{\gamma} = f_{\alpha}^{\beta} \cdot f_{\beta}^{\gamma} \)
- We call the \( f_{\alpha}^{\beta} \)’s bonds (or bonding maps).
An inverse limit of \( \{X_{\alpha} , f_{\alpha}^{\beta}, A \} \) is an object X of \( \mathcal{C} \) satisfying \( \forall \alpha \in A \) a morphism \( p_{\alpha} : X \to X_{\alpha} \) such that
(i) if \( \alpha \leq \beta \) then \( p_{\alpha} = f_{\alpha}^{\beta} \cdot p_{\beta} \)
(ii) If Z an object of \( \mathcal {C} \) satisfies: \( \forall \alpha \in A \) there exists \(q_ {\alpha} : Z \to X_{\alpha} \) such that \( q_{\alpha} = f_{\alpha}^{\beta} \cdot q_{\beta} \forall \alpha \leq \beta \) then there exists unique morphism \( q : Z \to X \) such that \( p_{\alpha} \cdot q = q_{\alpha} \forall \alpha \in A \)
Ex. If X as above exists, then it is unique up to isomorphism in \( \mathcal{C} \)