Is Čech cohomology homotopy invariant? I haven't been able to find a reference for a proof to this result if one exists so I'd love to be pointed in the direction of one.
The answer to this following question relies on this result for example:
Čech cohomology of a contractible space
EDIT:
I am referring to this definition of Čech cohomology:
For a closed topological subspace $A \subset X$ the Čech cohomology is the direct limit of the cohomology of open sets in $X$ containing $A$ - i.e. $\check{H}^n(A) := \varinjlim_{U \in \mathcal{U}_A} H^n(U)$.
or similarly for a closed topological pair:
$\check{H}^n(A,B) := \varinjlim_{(U,V) \in \mathcal{U}_{A,B}} H^n(U,V)$
 A: You do not make explicit which definition of Čech cohomology you are using. I guess it is the "classic" one based on nerves of open coverings. Here are two references giving proofs of the homotopy invariance of Čech cohomology:

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*The timeless classic from 1952


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*Eilenberg, Samuel, and Norman Steenrod. Foundations of algebraic
topology. Vol. 2193. Princeton University Press, 2015.

Have a look at Chapter IX "The Čech homology theory":

In this chapter the Cech homology and cohomology theories are
defined and the axioms for such are verified.

The homotopy axiom is stated as Theorem IX 5.1.


*Another classic from 1967


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*Spanier, Edwin H. Algebraic topology. Springer Science & Business Media, 1989.

Have a look at Chapter 6 "General cohomology theory and duality".
In section 4 the Alexander cohomolgy theory is introduced and the homotopoy axiom for Alexander theory is proved in section 5 (Theorem 6). In section 7 Čech cohomology theory is introduced (with coefficients in a presheaf; the constant presheaf gives coefficients in a module $G$). Spanier does not verify the axioms directly from the definition, but proves that Čech cohomology and Alexander cohomology agree for paracompact spaces (see section 8 Corollary 8). In particular, this proves the homotopy invariance of Čech cohomology.
