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# 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|MATH3062 Comparison of Hausdorﬀﬀ and topological dimension

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## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|Comparison of Hausdorﬀﬀ and topological dimension

a. Comparison of Hausdorff and topological dimension. One difference in the two dimensions we have defined is immediately apparent; the topological dimension is always an integer, while the Hausdorff dimension has no a priori reason to take integer values. Indeed, it can take any non-negative real value (see Exercise 2.20).

Another difference becomes apparent if we look at what notions are used in the definitions; the topological dimension can be defined for any topological space, whether or not it has a metric, while the Hausdorff dimension requires a metric for its definition. If we need to explicitly indicate the metric being used, we will write the Hausdorff dimension of $Z$ with respect to the metric $d$ as $\operatorname{dim}_H^d Z$.

This distinction becomes important when we observe that a single topological space can be equipped with multiple metrics. For example, the usual metric on $\mathbb{R}^d$ is given by Pythagoras’ formula
$$d(x, y)=\sqrt{\sum_i\left(x_i-y_i\right)^2},$$
but other metrics may be introduced by the formulae
\begin{aligned} \rho(x, y) &=\sum_i\left|x_i-y_i\right|, \ \sigma(x, y) &=\max _i\left|x_i-y_i\right|, \end{aligned}
and it is not hard to check that these metrics all induce the same topology on $\mathbb{R}^d$ (see Exercise 2.5). In particular, they all lead to the same topological dimension; do they all lead to the same Hausdorff dimension? To answer this question, we need some new definitions, giving three senses in which two metrics $d_1$ and $d_2$ on $\mathbb{R}^d$ (or more generally, on any metric space $X$ ) may be said to be “the same”.

## 数学代写|分形几何和混沌系统代考Fractal Geometry & Chaotic Dynamics代写|Metrics and topologies

b. Metrics and topologies. So far, all our examples of topological spaces have been metric spaces as well. One may rightly ask, then, if every example arises this way; given a topological space $(X, \mathcal{T})$, can we always find a metric $d$ on $X$ such that the sets in $\mathcal{T}$ are precisely those sets which are unions of $d$-balls? Such a space is called metrisable, and so we may ask, are all topological spaces metrisable?
It turns out that the answer is “no”: Some topologies do not come from metrics. But which ones? Given a particular topology, how can we tell whether or not it comes from a metric? To answer this question, we examine properties of metric spaces which do not follow from the axioms of a topological space.

Exercise 2.10. Let $(X, d)$ be a metric space, and fix $x \in X$. Show that the set ${x}$ is closed.

Exercise 2.11. Let $(X, d)$ be a metric space, and fix $x, y \in X$. Show that there exist disjoint open sets $U, V \subset X$ such that $x \in U$ and $y \in V$; that is, metric spaces are Hausdorff.

Exercise 2.12. Let $(X, d)$ be a metric space, and let $A, B \subset X$ be disjoint closed sets. Show that there exist disjoint open sets $U, V \subset X$ such that $A \subset U$ and $B \subset V$; that is, metric spaces are normal.

# 分形几何和混沌系统代考

## 数学代写|分形几何和混沌系统代考Fractal Geometry \& Chaotic Dynamics代 写|Comparison of Hausdorffff and topological dimension

$$d(x, y)=\sqrt{\sum_i\left(x_i-y_i\right)^2},$$

$$\rho(x, y)=\sum_i\left|x_i-y_i\right|, \sigma(x, y) \quad=\max _i\left|x_i-y_i\right|$$

## 数学代写|分形几何和混沌系统代考Fractal Geometry \& Chaotic Dynamics代 与|Metrics and topologies

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## MATLAB代写

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