TITLE: Hatcher 2.2 Exercise 33
QUESTION [6 upvotes]: The following is a question from Hatcher's "Algebraic Topology":

Let $X$ be a space such that $X$ is the union of $n$ open sets $A_i$ with the property that every intersection $A_{i_1}\cap \dots \cap A_{i_k}$ is either empty or has trivial reduced homology. Show that $\tilde H_p(X)=0$ for $p\geq n-1$. Give an example for each $n$ that shows this is the best possible bound.

I had no trouble proving the first statement with a little induction and an easy application of the Mayer Vietoris sequence. I am having a great deal of trouble thinking of an example that shows it is the best bound possible, however
For each $n$, I must find a space $X$ which can be decomposed into $n$ open sets such that all possible intersections of these sets are either empty or have trivial reduced homology. In particular, these open sets must themselves have trivial reduced homology, if I am understanding the question correctly. Furthermore, I must have $H_{n-2}(X)\neq 0$.
I am having trouble even coming up with a space that has trivial reduced homology in all dimensions. I was thinking that $\mathbb{RP}^n$ might work somehow with its decomposition into copies of projective spaces of lower dimension, but these aren't open sets.
Any hints?

REPLY [10 votes]: Consider the boundary of an $(n-1)$-simplex, which is homeomorphic to $S^{n-2}$. It has $n$ $n-2$-dimensional faces. Let your open sets be small open neighborhoods of these faces. Then the nonempty intersections will be neighborhoods of smaller dimensional faces, and will be contractible.