For any given topological space $X$ does $\chi(X)>2 \Rightarrow $ more than 1 connected component?
If not, when does it. And if true, can someone point me towards a proof. Thanks
For any given topological space $X$ does $\chi(X)>2 \Rightarrow $ more than 1 connected component?
If not, when does it. And if true, can someone point me towards a proof. Thanks
Certainly not. For example, $\chi(\mathbb CP^n)=n+1$, but of course $\mathbb CP^n$ is connected.
It is true for surfaces, though: Euler char of any connected surface $\le2$ (this follows, for example, from the classification of surfaces).