equivalence relations on a subset of k dimensional euclidean space

Question

Problem: Let a and b be points in $\mathbb{R}^k$. A path from a to b is a continous function on the unit interval $[0, 1]$ with values in $\mathbb{R}^k$, a function $X: [0, 1]$, sending t to $X(t)=(x_1(1), \cdots x_k(t))$, such that $X(0)=a$ and $X(1)=b$. If $S$ is a subset of $\mathbb{R}^k$ and if a and b are in S, define $a$ ~ $b$ iff a and b can be joined by a path lying entirely in S. Prove that ~ is an equivalence relation on $S$.

This problem is from Artin's algebra book.

I am able to show reflexity and symmetry ($X(1-t)$) but don't know how to show transitivity....

If $a~b$ by $X(t)$ and $b~c$ by $Y(t)$ then $Z(t)= X(t)+Y(t)-b$ seems good except for...I don't if $Z$ will be in $S$ or not. if $k=1$ then by IVT we are done...but I dont know how to prove for $R^k$
I don't know any Topology.

Answer 1

I think you can take

$Z(t) = X(2t)$ for $t \in [0,0.5]$

$Z(t) = Y(2t-1)$ for $t \in ]0.5,1]$

Since $X(0.5)=Y(0.5)=b$, $Z$ is continuous and in $S$

Answer 2

To show transitivity of this relation is to show: For $a,b,c\in S$ one has

$$\text{If $a$ can be connected to $b$ and $b$ can be connected to $c$, show that $a$ can be connected to $c$.}$$

Note that we assume that there is a continuous path from $a$ to $b$ and a continuous path from $b$ to $C$, which lie entirely in $S$. Just concatenate these (and maybe rescale) to ge a continuous path from $a$ to $c$, which lies in $S$.