Topologies in $\mathbb{R}^2$

In the following applet differerent topologies in $\mathbb{R}^2$ are visualized. The orange shape corresponds to an open set in the given topology.

Select one topology of the following list:

Metric topologies

Let $d: X \times X \rightarrow \mathbb{R}$ be a metric, i.e. Metric 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 triangle inequality implies that this is equivalent to: $$\forall x\in O \,\exists \varepsilon>0: B_{\varepsilon}(x) \subset O$$.


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.

Box topology

A basis of the box topology in $\mathbb{R} \times \mathbb{R}$ can be formed by sets $B_1 \times B_2$ where $B_1$, $B_2$ are basis elements of the standard topology in $\mathbb{R}$. In finite $\mathbb{R}$ vectorspaces, this construction gives the same topology as the one that is induced by the $\ell$-norms

Discrete topology

Any set in this topology is an open set. The set of singletons can be considered as basis.