So I've been working through this book (A Basic Course in Algebraic Topology, by William Massey) in preparation for an Algebraic Topology course I'm going to take soon, and I ran into trouble with this exercise.
Let ${U_i}$ be an open covering of the space X having the following properties:
(a) There exists a point $x_0$ such that $x_0$ ϵ $U_i$ for all i.
(b) Each $U_i$ is simply connected.
(c) If i≠j, then $U_i$ ∩ $U_j$ is arcwise connected.
Prove that X is simply connected.
The book hints at proving that any loop f: I$\to$X (where I = [0,1] ) based at $x_0$ is trivial. I know I should use the fact that {$f^{-1}$($U_i$)} is an open covering of the compact space I, which means that there exists a finite subcovering of I. However, I am not sure how to proceed from there. Any hint at all would be appreciated. Thanks in advance.
Take a path $\phi\colon I \to X$ starting and ending at $x_0$. From the Lebesgue number lemma (Proof of the Lebesgue number lemma) for the covering $\phi^{-1}(U_i)$ of $I$ we conclude that there exists $n \ne 1$ so that $[\frac{k-1}{n}, \frac{k}{n}]$ is inside some $\phi^{-1}(U_i)$, that is, $\phi([\frac{k-1}{n}, \frac{k}{n}] )\subset U_{i_k}$. For $1< k <n$ , $k$ natural, the point $\phi(\frac{k}{n})$ is in $U_{i_k}\cap U_{i_{k+1}}$, and so is $x_0$. Take a path $\psi_k \colon I \to U_{i_k}\cap U_{i_{k+1}}$ connecting $x_0$ and $\phi(\frac{k}{n})$.
Here are the steps :
Decompose $\phi$ into a composition of $n$ paths $\phi_k$ that are basically the restrictions of $\phi$ to $[\frac{k-1}{n}, \frac{k}{n}]$
Show that
$$\phi_1 \star \ldots \star \phi_n \simeq \phi_1 \star \psi_1^{-1} \star \psi_1 \star \phi_2 \star \psi_2^{-1} \ldots \star \phi_n $$
For each $k$ the path $\psi_{k-1}\star \phi_k \star \psi_k$ is a closed look in $U_{i_k}$ and therefore homotopic to a trivial path.
Piece together to get a homotopy from the loop $\phi$ to the constant loop at $x_0$.
There are some alternate, equivalent ways to define the path composition and the equivalence of paths, some making certain proofs easier -- compare with the path composition in the book by Crowell and Fox, Knot Theory http://www.maths.ed.ac.uk/~aar/papers/crowfox.pdf.