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Suppose we have a (co)filtered digaram $\dots \rightarrow X_{2}\rightarrow X_{1}$ of topological space. Is is true that the natural map $\pi_{0}[\lim X_{i}]\rightarrow \lim \pi_{0}(X_{i})$ is an isomo …

Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?

I think I understand how to define the algebraic fundamental group $\pi^{alg}_{1}(X)$ of a scheme and I think I understand the relation between $\pi^{alg}_{1}(X)$ and $\pi_{1}(X(\mathbb{C}))$, where …

I'm not sure about the following result: Theorem (?): Let $f_{\bullet,\bullet}: X_{\bullet,\bullet}\rightarrow Y_{\bullet,\bullet}$ a map of bisimplicial sets such that for any natural number $n$, …

If you want to prove that group cohomology $H^{\ast}(\Sigma_{n},\mathbb{Q})=0$ you use the fact that $\mathbb{Q}$ is a projective $\mathbb{Q}[\Sigma_{n}]$-module (Maschke’s theorem) therefore by defin …