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Warmup: Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric realizati …

The nLab page on partitions of unity mentions the application of partitions of unity as a way to construct continuous maps to geometric realizations of simplicial spaces. However I often feel uncomfor …

Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms (equivariant bu …

I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$. Milnor's On spaces having the homotopy type …

I asked this question on math.stackexchange a week ago, but did not get an answer. First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situ …

I asked this question on math.stackexchange, but did not get an answer. Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't …