2
votes
1answer
73 views

Behavior of Hausdorff dimension under homeomorphisms

Let $X$ and $Y$ be metric spaces, $f : X\rightarrow Y$ a homeomorphism. Denote by $\dim_{\mathcal H}$ the Hausdorff dimension. I know that it is possible that $\dim_{\mathcal H} Y < \dim_{\mathcal ...
0
votes
1answer
133 views

Hausdorff Measure and Hausdorff Dimension

Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension? The Hausdorff Measure is defined as the following: Let $(X,d)$ be a metric space. $\forall S \subset X$, ...
1
vote
1answer
236 views

Hausdorff dimension of the set of rational numbers within a certain interval?

Intro: The Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. In general the Hausdorff dimension ...
4
votes
2answers
110 views

A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
3
votes
0answers
82 views

Lipschitz continuity for an iterated function system

Let $(M,d_M)$ and $(N,d_N)$ be metric and $$ CB(M)=\{\mbox{all closed bounded subsets of }M\}. $$ Let $f: M\to N$ be a Lipschitz map with Lipschitz constant $L$. Define a map $$ F:(CB(M),\rho)\to ...
0
votes
1answer
220 views

Finite sets are dense with respect to Hausdorff distance

Let $(X,d)$ be a complete metric space and consider \begin{align*} BC(X)&= \lbrace C\subset X\;|\;C\neq\emptyset\text {, closed and bounded} \rbrace\cr \mathrm{Fin}(X)&= \lbrace ...
3
votes
0answers
184 views

Is the Hausdorff semi-distance Lipschitz?

Let $X$ be Banach (with metric $d$) and let $H(X)$ be the set of closed bounded subsets of $X$. Define for $A,B\in H(X)$ $$\delta(A,B)=\sup_{a\in A}\inf_{b\in B}d(a,b)$$ be the Hausdorff semi-distance ...