Why is the topological space of reals defined to be path connected?

Question

Note: $R$ is the topological space of reals, equipped with the standard topology.

A set $X$ is said to be path connected if there exists a continuous map, $f:R \to X$, such that for any $r_1,r_2 \in R$, with $r_1<r_2$, we have $f(r_1) = x$ and $f(r_2) = y$ and $x,y \in X$.

This seems like we are defining the topology on reals, $R$ to be path connected, by definition, as there always exists a homomorphism $f:R \to R$ and we are trying to extend this notion to other topological spaces $X$, through the continuous map $f:R \to X$.

It seems like, we could consider any connected space $Y$ and define it to be path connected. Then call $X$ to be path connected if there exists a continuous map $f:Y \to X$ satisfying the above conditions.

It seems like we are interested only in the topologies which are homomorphic to $R$. Why is a special preference given to the topology on reals?

Edit: It was pointed out by Jose Carlos Santos that my definition of path connectedness is wrong.

Answer 1

Your definition of path-connected is worst than wrong: it desn't make sense. How can we have $f(r_1)=x$ and $f(r_2)=y$ for any $r_1,r_2\in\mathbb R$? Even assuming that $r_1\neq r_2$.

The usual definition of path-conneced is: $X$ is path-connected if, for any $x,x'\in X$, there is a continuous map from some interval $[a,b]$ of $\mathbb R$ (with $a<b$) into $X$ such that $f(a)=x$ and that $f(b)=x'$. But $f$ usually is not a homeomorphism. Acutally, it doesn't even have to be a homeomorphism from $[a,b]$ onto $f\bigl([a,b]\bigr)$, since $f$ might not be one-to-one.

Answer 2

(Concerning your definition of path connected see José Carlos Santo's answer.)

"Path connected" is the most intuitive description of connectedness, and is easy to work with, e.g., it allows to pull back functions defined on a large and "complicated" space to a real interval. A large part of spaces and domains occurring in analysis or geometry are in fact path connected.

Of course there are situations where we have to recurse to the more subtle "official" description of connectedness, e.g., when dealing with the set $$S:=\left\{(x,y)\>\biggm|\>x\ne0\ \wedge\ y=\sin{1\over x}\right\}\cup \bigl(\{0\}\times[{-1},1]\bigr)\subset{\mathbb R}^2\ .$$ The "official" definition, when it has to be applied to relative spaces, as in the above example, can be quite delicate. Therefore we are glad when a set under study is not only connected but even path connected.