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

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

Definition 3.65 A topological space is called a Hausdorff space, or T2 space, if any two distinct points admit disjoint neighbourhoods.

In other words, given distinct points $x, y$ in a Hausdorff space $X$ there exist neighbourhoods $U \in \mathcal{I}(x)$ and $V \in \mathcal{I}(y)$ such that $U \cap V=\emptyset$.

Not all spaces are Hausdorff: the trivial topology is not T2, except for the irrelevant cases where the space is empty or has only one element.

Example 3.66 Any metric space is Hausdorff. If $d$ is the distance and $x \neq y$, then $d(x, y)>0$. When $0<r<\frac{d(x, y)}{2}$ the balls $B(x, r)$ and $B(y, r)$ are disjoint, simply because if there was a $z \in B(x, r) \cap B(y, r)$, the triangle inequality would imply $d(x, y) \leq d(x, z)+d(z, y)<2 r<d(x, y)$.

Non-metrisable Hausdorff spaces do exist: Exercises 3.61 and 3.62 are but two examples.

## 数学代写|拓扑学代写TOPOLOGY代考|Connectedness

It’s part of the human intuition to answer ‘two’ when asked ‘How many pieces make up the space $X=\mathbb{R}-{0}$ ?’ This happens because we are able to distinguish two parts in $X$, namely $X_{-}={x<0}$ and $X_{+}={x>0}$, that in our mind match the word ‘piece’ of the question.

Topological structures allow to define in a mathematically precise way the notions of connected space and connected component, that correspond to the naïve concepts of ‘one single piece’ and ‘piece of cake’ if the cake is already sliced.

Definition 4.1 A topological space $X$ is called connected if the only subsets that are both open and closed are $\emptyset$ and $X$. A non-connected topological space is called disconnected.
Lemma 4.2 On a topological space $X$ the following properties are equivalent:

1. $X$ is disconnected;
2. $X$ is the disjoint union of two open, proper subsets;
3. $X$ is the disjoint union of two closed, proper subsets.
Proof (1) $\Rightarrow$ (2), (3). Let $A \subset X$ be open, closed and non-empty. If $A \neq X$ the complement $B=X-A$ is open, closed and non-empty and $X$ is the disjoint union of $A$ and $B$.
(2) $\Rightarrow$ (1). Suppose $A_1 \cup A_2=X$ with $A_1, A_2$ open, non-empty and disjoint. Then $A_1=X-A_2$ is closed, too.
(3) $\Rightarrow$ (1). If $C_1 \cup C_2=X$, where $C_1, C_2$ are closed, non-empty and disjoint, $C_1=X-C_2$ is open.

## 数学代写|拓扑学代写TOPOLOGY代考|Connectedness

$X$ 断开连接;

$X$ 是两个开的固有子集的不相交并;

$X$ 是两个封闭的固有子集的不相交并。

(2) $\Rightarrow$(1).设$A_1 \cup A_2=X$, $A_1, A_2$是开的、不空的、不相交的。然后$A_1=X-A_2$也关闭了。
(3) $\Rightarrow$(1)若$C_1 \cup C_2=X$，其中$C_1, C_2$为闭合、不空、不接合，则$C_1=X-C_2$为打开。

## MATLAB代写

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