I have the following space $X=[0,1) \cup \{y,z\}$ with $y \neq z$ and $y,z \notin [0,1)$.
Where the basic open for point in $[0,1)$ are taken as the basics to the topology that R inherits to the space.
The basics opens for $y$ are the sets of the form $\{y\} \cup (b,1)$, where $b \in (0,1)$.
And the basic open for $z$ are the sets of the form $\{z\} \cup (b,1)$, where $b \in (0,1)$.
I need to prove that $X$ is path connected. My definition of path-connected is that for any two points $p,q \in X$, there exists $f:[0,1] \rightarrow X$ continuous such that $f(0)=p$ and $f(1)=q$.
Could someone help me find this function?
Thanks you.
Hint : Prove that :
for any $x \in [0,1)$, the path $\gamma_x : [0,1] \rightarrow X$ defined for all $t \in [0,1]$ by $\gamma_x(t)=tx$ is a continuous path joining $0$ to $x$,
the path $\gamma_y: [0,1] \rightarrow X$ defined by
$$\forall t \in [0,1), \ \gamma_y(t)=t \quad \text{and} \quad \gamma_y(1)=y$$ $\quad\ $ is a continuous path joining $0$ to $y$,
$$\forall t \in [0,1), \ \gamma_z(t)=t \quad \text{and} \quad \gamma_z(1)=z$$ $\quad\ $ is a continuous path joining $0$ to $z$.