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.

You can select a metric from the following list:

Or you can enter your own metric (in CindyScript):

Explaination on Metrices

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


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.


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


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