Random continuous functions on metric induced topologies on $\mathbb{R}^2$

Let $\mathcal{T}$ be the topology induced by a metric $d$ on $\mathbb{R}^2$. Then a random continuous function $$f: (\mathbb{R}^2, \mathcal{T}) \rightarrow (\mathbb{R}, \text{standard topology})$$ can give an idea about the "shape" of the topological space $(\mathbb{R}^2, \mathcal{T})$: If $f(x)$ and $f(y)$ differ, then $x$ and $y$ cannot be arbitary "close".

Generation of random continuous functions $f: (\mathbb{R}^2, \mathcal{T}) \rightarrow (\mathbb{R}, \text{standard topology})$

Let ${(x_i)}_{i\in \mathbb{N}}$ be a random sequence in $\mathbb{R}^2$. For a given $x_i$ the function $$\mathbb{R}^2 \rightarrow \mathbb{R}, x \mapsto d(x_i, x)$$ is continuous for the topology on $\mathbb{R}^2$ that is induced by the metric $d$. Since $x \mapsto \frac{1}{d(x_i, x)^2 + 1}$ is both continuous and bounded by $1$, the sum in $$ f(x) := \sin\left(\sum_{i=0}^{\infty} (-0.995)^i \frac{1}{d(x_i, x)^2 + 1}\right) $$ converges uniformly and gives raise to a random continuous function $f: (\mathbb{R}^2, \mathcal{T}) \rightarrow (\mathbb{R}, \text{standard topology})$. In the visualization, an approximate $f$ is computed for an arbitrary metric $d:\mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}$. A pixel $x$ is colored with the value of $f(x)$ where $-1$ will displayed as black and $+1$ as white.

SCNF-metric Jungle-River-metric standard-metric bounded standard-metric Manhatten-metric metric induced by max-norm disjoint rings

Explaination on Metrices

$d: X \times X \rightarrow \mathbb{R}$ is a metric if

• $ d(a,b)\geq 0 $ for all $a \neq b$ non-negativity
• $ d(a,b)=0\Leftrightarrow a=b $
• $d(a,b)=d(b,a)$ symmetry
• $d(a,c)\leq d(a,b)+d(b,c)$ triangle inequality

Metrices induced topologies have the open balls $B_r(a) = \{ b \in X \mid d(a,b) < r \}$ as basis, i.e. every open set $O$ can be written as an (probably infinite) union of balls.

Standard-Metric

The standard-metric is induced by the $\ell_{2}$-norm, i.e. $$d(a,b) = ||a-b||_2 = \sqrt{(a_x-b_x)^2+(a_y-b_y)^2}$$

Manhattan distance

THe Manhatten metric is induced by the $\ell_{1}$-norm, i.e. $$d(a,b) = ||a-b||_1 = |a_x-b_x|+|a_y-b_y|$$ The name is motivated by the grid layout of most streets on the island of Manhattan. The Manhatten metric induces the same topology as the standard metric.

SCNF-metric

If you want to travel by train in France, it is very likely that the fastest connection goes through Paris. In $\mathbb{R}^2$ this can be modelled as:

• $d(a,b)=||a-b||_2$ if $x$ and $y$ have the same phase and
• $d(a,b)=||a||_2 + ||y||_2$
in all other cases (i.e. passing through $0$ = Paris)

The induced topological space corresponds to an uncountable a set of disjoint rays joined at a point.

Jungle-River-metric

Here, we assume that one can only travel on vertical lines and the x-axis.

• $d(a,b)=|a_y-b_y|$ if $a_x=b_x$ and
• $d(a,b)=|a_y| + |a_x-b_x| + |b_y|$
in all other cases (i.e. we are traveling on the x-axis)