If two topological spaces are weak homotopy equivalent to each other, are their Cech cohomology groups the same?



Weak homotopy equivalence and Cech cohomology
$$T = \left\{ \left( x, \sin \frac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,y)\mid y\in[-1,1]\}$$


This has trivial homotopy groups in degrees $\ge1$ but according to Wikipedia nontrivial Čech cohomology in degree 1.


