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.

Topology induced by standard-metric Topology induced by Manhatten-metric Topology induced by SCNF-metric Topology induced by Jungle-River-metric Box topology Discrete topology

Metric topologies

Let $d: X \times X \rightarrow \mathbb{R}$ be a metric, i.e.

• $ 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

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: $$\forall x\in O \,\exists \varepsilon>0: B_{\varepsilon}(x) \subset O$$.

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)

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.