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# 数学代写|拓扑学代写TOPOLOGY代考|MATH271 Abstract Topological Spaces

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## 数学代写|拓扑学代写TOPOLOGY代考|Abstract Topological Spaces

Roughly speaking, a topological space is a set together with some notion of what it means for a subset to be considered open. The precise definition is as follows:

Definition 3.13 A topological space is a set $S$ together with a collection $\mathscr{T}$ of subsets of $S$ (called the open sets in $S$ ) so that
$\varnothing, S \in \mathscr{T}$.

• If $A$ is any set and $\left{S_\alpha\right}_{\alpha \in A}$ is a collection of subsets of $S$ indexed by $A$, so that each $S_\alpha \in \mathscr{T}$, then $\bigcup_{\alpha \in A} S_\alpha \in \mathscr{T}$.
• If $S_1, S_2, \ldots, S_n \in \mathscr{T}$, then $\bigcap_{i=1}^n S_i \in \mathscr{T}$.
We call $\mathscr{T}$ a topology on $S$.
Remark 3.14 It is possible to put many different topologies on a set $S$. In particular, if $S=\mathbb{R}^n$, then the open sets of some exotic topology need not satisfy the ball property that we discussed in Chapter 1.

When we translate these into statements about open sets, these properties are saying the following:

• The empty set and all of $S$ are open sets.
• The union of any collection of open sets is open.
• The intersection of a finite collection of open sets is open.

## 数学代写|拓扑学代写TOPOLOGY代考|The Quotient Topology

One particularly important type of topology is called the quotient topology, which is just the thing we need for ID spaces.

Definition 3.15 Let $S$ be a topological space with topology $\mathscr{T}$, and let $\sim$ be an equivalence relation on $S$. We define a topology, called the quotient topology, on the set $S / \sim$ of equivalence classes modulo $\sim$ as follows: let $p: S \rightarrow S / \sim$ be the map that takes an element of $S$ to its equivalence class modulo $\sim$. Then we define a set $U \subset S / \sim$ to be open if $p^{-1}(U) \in \mathscr{T}$, i.e. if $p^{-1}(U)$ is an open set of $S$.

This is relevant for ID spaces, because they are defined to be sets of equivalence classes. For example, we can view the torus, in its ID space form, as being the square $[0,1] \times[0,1]$ of its ID space, modulo the equivalence relation that points on the left side are equivalent to points on the right side, and similarly with top and bottom sides. To set this up as an equivalence relation, we declare that $(a, 0) \sim(a, 1)$ and $(0, b) \sim$ $(1, b)$. The only other equivalences we allow are the trivial ones $(a, b) \sim(a, b)$. For $(a, b)$ in the interior $(0,1) \times(0,1)$ of the square, a small disk in $(0,1) \times(0,1)$ centered at $(a, b)$ is an open neighborhood of $(a, b)$. But for $(a, b)$ on an edge or vertex of a square, a neighborhood looks a little different, as shown (in the case of an edge) in Figure 3.12.

Similarly, we can view a circle as a quotient of the closed interval $[0,1]$, by saying that $0 \sim 1$, and the only other equivalences are $a \sim a$ for $a \in[0,1]$.

## 数学代写|拓扑学代写TOPOLOGY代考|Abstract Topological Spaces

• 如果 $A$ 是任何集合并且 $\backslash$ left 缺少或无法识别的分隔符
是子集的集合 $S$ 被索引 $A$, 这样 每个 $S_\alpha \in \mathscr{T} ，$ 然后 $\bigcup_{\alpha \in A} S_\alpha \in \mathscr{T}$.
• 如果 $S_1, S_2, \ldots, S_n \in \mathscr{T} ，$ 然后 $\bigcap_{i=1}^n S_i \in \mathscr{T}$. 我们称之为 $\mathscr{T}$ 上的拓扑 $S$.
备注 $3.14$ 可以将许多不同的拓扑放在一个集合上 $S$. 特别是，如果 $S=\mathbb{R}^n$ ，那么一些奇异拓扑的开集不 需要满足我们在第 1 章中讨论的球性质。
当我们将这些转化为关于开集的陈述时，这些属性表示以下内容:
• 空集和所有 $S$ 是开集。
• 任何开集集合的并集都是开集。
• 开集的有限集合的交集是开的。

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