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

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

Definition 3.1 A topology on a set $X$ is a family $\mathcal{T}$ of subsets of $X$, called open sets, satisfying the following requirements:

(A1) $\emptyset$ and $X$ are open sets;
(A2) the union of any number of open sets is an open set;
(A3) the intersection of two open sets is an open set.
A set equipped with a topology is called a topological space. Its elements are called points.

Condition $\mathrm{A} 3$ implies that any finite intersection of open sets is still an open set: in fact if $A_1, \ldots, A_n$ are open sets we can write $A_1 \cap \cdots \cap A_n=\left(A_1 \cap \cdots \cap A_{n-1}\right) \cap A_n$. By induction on $n$ the set $A_1 \cap \cdots \cap A_{n-1}$ is open, so by $\mathrm{A} 3$ also $A_1 \cap \cdots \cap A_n$ is open.

Any set admits at least one topology, typically several. For instance the family $\mathcal{T}=\mathcal{P}(X)$ of subsets of $X$ is a topology called the discrete topology, while the family $\mathcal{T}$ containing only the empty set and $X$ is a topology called trivial or indiscrete.

## 数学代写|拓扑学代写TOPOLOGY代考|Interior of a Set, Closure and Neighbourhoods

The concept of adherence defined in Sect. 1.3 generalises to arbitrary topological structures.

Definition 3.12 Given a subset $B$ in a topological space $X$, one writes $\bar{B}$ for the intersection of all closed sets containing $B$ :
$$\bar{B}=\bigcap{C \mid B \subset C \subset X, C \text { closed }} .$$
Equivalently, $\bar{B}$ is the smallest closed set in $X$ containing $B$, and is called the closure of $B$; its points are said to be adherent to $B$.

Example 3.13 If $A \subset B$, any closed set containing $B$ also contains $A$, and so $\bar{A} \subset \bar{B}$. If $A_i$ is a family of subsets in a topological space, then $\cup_i \overline{A_i} \subset \overline{\cup_i A_i}:$ as $A_j \subset \cup_i A_i$, in fact, $\overline{A_j} \subset \overline{\cup_i A_i}$ for any $j$.

Definition 3.14 A subset $A$ in a topological space $X$ is said to be dense if $\bar{A}=X$ or, equivalently, if $A$ meets every non-empty open subset of $X$. More generally, if $A \subset B$ are subsets, $A$ is dense in $B$ when $B \subset \bar{A}$.

Example 3.15 1. In a space with the trivial topology any non-empty subset is dense.

1. In a space with the discrete topology no proper subset is dense.
2. The set of rationals is dense in the space $\mathbb{R}$ equipped with the Euclidean topology.
3. In a space with the cofinite topology any infinite subset is dense.
Closely related to the closure are the notions of interior and boundary.

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

(A1) $\emptyset$和$X$是开集;
(A2)任意数量的开集的并集是一个开集;
(A3)两个开集的交集是一个开集。

## 数学代写|拓扑学代写TOPOLOGY代考|Interior of a Set, Closure and Neighbourhoods

1.3节中定义的粘附概念可以推广到任意拓扑结构。

$$\bar{B}=\bigcap{C \mid B \subset C \subset X, C \text { closed }} .$$

3.14子集 $A$ 在拓扑空间中 $X$ 据说密度是多少 $\bar{A}=X$ 或者，等价地，如果 $A$ 满足的所有非空开放子集 $X$． 更一般地说，如果 $A \subset B$ 是子集， $A$ 是稠密的 $B$ 什么时候 $B \subset \bar{A}$．

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