# The Peano curve: continuity can be counter-intuitive

The Peano curve is obtained as the limit of piecewise-linear continuous (even closed) curves $\gamma_n$. Denote by $K=\{|x|+|y|\le 1\}$ the square (rotated by $\frac \pi/4$) and by $\mathbb Z^2=\{(x,y):x,y\in\mathbb Z$ the grid of horizontal and vertical lines at distance 1 from each other, then one can construct a family of piecewise-linear continuous curves $\gamma_n:[0,1]\to\mathbb R^2$ which visits all points of the intersection $K\cap\frac1{2^n}\mathbb Z^2$ in such a way that $|\gamma_n(t)-\gamma_n(t)|<\frac1{2^n}$ uniformly on $t\in[0,1]$.

This sequence of curves converges uniformly to a function (curve) $\gamma_*:[0,1]\to\mathbb R^2$ and this curve is closed and continuous for the same reasons that justify continuity of the Koch snowflake curve.

What are the properties of the images $C_n=\gamma_n([0,1])$ and of the limit curve $C_*=\gamma_*([0,1])$?

• Each curve $C_n$ for any finite $n$ is piecewise-linear. It has zero area in the sense that for any $\varepsilon > 0$ the curve $C_n$ can be covered by a finite union of (open) rectangles with the total area less than $\varepsilon$;
• Each curve $C_n$ has finite length (although it grows to infinity as $n\to\infty$, – check it!).
• The limit curve $C_*$ has no length (that’s the same as saying that it has infinite length). Moreover, unlike many other curves of infinite length (say, the straight line $\{y=0\}\subseteq\mathbb R^2$), no part $\gamma([a,b]),\ a, of $C_*$ has finite length!
• The limit curve $C_*$ coincides with the square $K$, hence fills the area equal to 2.

All these assertions are easy except for the last one. Let’s prove it.

Consider the images $C_n=K\cap \frac1{2^n}\mathbb Z^2$. The union of these images is dense in $K$: by definition, this means that any point $P\in K$ can be approximated by a sequence of points $P_n\in C_n$ which converge to $P$ as $n\to\infty$. Being in the image of $\gamma_n([0,1])$, each point $P_n$ is the image of some point in [0,1]: $\exists a_n\in[0,1]:\ \gamma(a_n)=P_n$. Such point may well be non-unique, and in any case we have absolutely no knowledge of how the points $a_1,a_2,\dots$ are distributed over [0,1].

However, we know that the sequence $a_n\in [0,1]$ must have an accumulation point $a_*\in [0,1]$, which is by definition a limit of some infinite subsequence. (This won’t be the case if instead of [0,1] we were dealing with the curves defined on the entire real line!). Replacing the sequence by this subsequence, we see that it still converges to the same limit, $P_n=\gamma(a_n)\to a_*=\gamma_*(a_*)=P$. Thus we proved that an arbitrary point in $K$ lies in the image: $P\in C_*$.

# Topology: the study of properties preserved by continuous maps (functions, applications, …)

Definition. A neighborhood of a point $a\in\mathbb R^n$ in the Euclidean space is any set of the form $\{x:|x-a| 0$, where $| ??? |$ is a distance function satisfying the triangle inequality. Examples:

• $|x|=\sqrt{x_1^2+\cdots+x_n^2}$ (the usual Euclidean distance on the line, on the plane, …) for $x=(x_1,\dots,x_n)\in\mathbb R^n$;
• $|x|=\max\{|x_1|, \dots, |x_n|\}$ (in the above notation);
• $|x|=|x_1|+\cdots+|x_n|$.

Definition. A subset $A\subset\mathbb R^n$ of the Euclidean space (OK, plane) is called open, if together with any its point $a\in A$ it contains some neighborhood of $a$.
A subset is called closed, if the limit of converging infinite sequence $\{a_n\}\subset A$ again belongs in $A$.

Theorem. A subset $A$ is open if and only if its complement $\mathbb R^n\smallsetminus A$ is closed.

Theorem. The union of any family (infinite or even uncountable) of open sets is open. Finite intersection of open sets is also open (for infinite intersections this is wrong).
Corollary. Intersection of any family (infinite or even uncountable) of closed sets is closed. Finite union of closed sets is also closed (for infinite intersections this is wrong).

One can immediately produce a lot of examples of open/closed subsets in $\mathbb R^n$. It turns out that any property that can be formulated using only these notions, is preserved by maps which are continuous together with their inverses. The corresponding area of math is called topology.