I am working on Exercise 4.1.11 in Hatcher's Algebraic Topology text, which is as follows:
Show that a CW complex $X$ is contractible if it is the union of an increasing sequence of subcomplexes $X_1 \subset X_2 \subset \cdots$ such that each inclusion $X_i \hookrightarrow X_{i+1}$ is nullhomotopic.
By applying Whitehead's Theorem, if we show that $\pi_n(X,x_0) = 0$ for all $n$ and any point $x_0 \in X$ and that $X$ is connected, it will follow that $X$ is contractible.
It is already shown here that $\pi_n(X,x_0) = 0$ for all $n$ and any point $x_0 \in X$. What's not shown there, however, is that $X$ is connected.
How can we show that $X$ is connected?
Thanks!
Best Answer
If you know that $\pi_0(X,x_0)=0$, then the result is trivial. Explicitly, you can consider two points $x,y$. There is an $i$ such that $x,y\in X_i$. Since the inclusion $X_i\to X_{i+1}$ is nullhomotopic, there is an arc connecting these two points in $X_{i+1}$.