End (topology)
In topology, an end is, informally, a way for a topological space to be "infinite." More formally, it describes a way to leave a space without ever returning. The notion of an end aims to capture the idea of the number of "holes at infinity" of a space.
Definition
Let $X$ be a topological space. Typically, to have a meaningful definition of ends, $X$ is required to have some additional properties. Commonly, $X$ is assumed to be locally compact, Hausdorff, and connected (and often also locally connected). A nested sequence of nonempty open sets ${U_i}{i=1}^{\infty}$ in $X$ is a sequence where $U{i+1} \subseteq U_i$ for all $i$. An end of $X$ is a nested sequence ${U_i}_{i=1}^{\infty}$ of nonempty open sets such that:
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Each $U_i$ has compact boundary.
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$\bigcap_{i=1}^{\infty} \overline{U_i} = \emptyset$, where $\overline{U_i}$ denotes the closure of $U_i$. This condition ensures that the end "goes off to infinity" and doesn't converge to a point in the space.
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For each i, $U_i$ is connected.
Two nested sequences ${U_i}{i=1}^{\infty}$ and ${V_i}{i=1}^{\infty}$ are considered equivalent if for every $i$, there exists a $j$ such that $V_j \subseteq U_i$, and for every $k$, there exists an $l$ such that $U_l \subseteq V_k$. An end of $X$ is then an equivalence class of such nested sequences under this equivalence relation.
Properties and Examples
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The number of ends of a space can be a topological invariant used to distinguish between spaces.
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The real line $\mathbb{R}$ has two ends, corresponding to going to $+\infty$ and $-\infty$.
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The Euclidean plane $\mathbb{R}^2$ has one end. To see this, consider complements of large closed disks centered at the origin.
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The integers $\mathbb{Z}$ (with the discrete topology) have infinitely many ends.
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The number of ends of a graph is an important concept in graph theory, especially infinite graph theory.
Applications
The concept of ends is used in:
- Topology: To study the large-scale structure of spaces.
- Geometric Group Theory: To understand the asymptotic behavior of groups via their Cayley graphs.
- Riemann Surface Theory: To analyze the behavior of functions near the "ideal boundary" of Riemann surfaces.
The notion of ends is a powerful tool for understanding the global properties of topological spaces, especially those that extend infinitely in some sense.