For any space $X$, let $Y=X\times I$, and topologize $Y$ by defining basic neighbourhoods of $(x,y)$ as -
- $(x,y), y\neq0: U_{(x,y)} = \{x\}\times B_\epsilon(y)$
- $(x,0): U_{(x,0)} = \{(x',z):z'\in U, 0\leq z < \epsilon_z \},$ where $U$ is a neighbourhood of $x$ in $X$, and $\epsilon_z > 0\ $ $\forall z\in U$
Let $X^*$ be the quotient space of $Y$ on identifying all points $(x,1), x\in X$.
Then, show that $X^*$ is path connected, and that every continuous function $f:X\to Z$ can be extended to continuous $F:X^*\to Z$, where $Z$ is path connected.
I've been able to show that $X^*$ is connected. However, I've failed to show the extension property. Any help would be appreciated!
HINT: Let $p$ be the point at the ‘top’. Pick any base point $x_0\in X$, and let $z_0=f(x_0)$. Let $F(\langle x_0,y\rangle=z_0$ for $y\in[0,1)$, and let $F(p)=z_0$ as well. For each $x\in X\setminus\{x_0\}$ there is a path in $Z$ from $f(x)$ to $z_0$; define $F$ on $\{x\}\times[0,1)$ so that it follows that path. To show continuity, use the fact that the sets $\{x\}\times(0,1)$ in $X^*$ are pairwise disjoint and open.