# Day 3 (notes from Craig’s Lecture)

#### 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 $$\mathcal{C}$$