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A sequence of inclusions whose colimit is not a homotopy colimit. (English) Zbl 1328.55010

Let \(X_1\hookrightarrow X_2\hookrightarrow\cdots\hookrightarrow X_n\hookrightarrow\cdots\) be a sequence of inclusions of topological spaces and consider the corresponding topological colimit \(\text{colim}_n X_n\) and the homotopy colimit \(\text{hocolim}_n X_n\) (which in this situation is the so-called mapping telescope). It is a well-known and widely used fact in homotopy theory that, if all spaces \(X_n\) satisfy the \(T_1\)-axiom, then the natural mapping \(h: \text{hocolim}_n X_n\to \text{colim}_n X_n\) is a weak homotopy equivalence.
Motivated by the observation that the same is true if instead all spaces \(X_n\) are assumed to be Alexandroff spaces, i.e., topological spaces in which every intersection of open sets is open, the authors wondered whether the conclusion holds without any restriction on the spaces \(X_n\). They answer their question by constructing a sequence \(X_1\hookrightarrow X_2\hookrightarrow\cdots\hookrightarrow X_n\hookrightarrow\cdots\) of inclusions such that the natural mapping \(h:\text{hocolim}_n X_n\to \text{colim}_n X_n\) is not a weak homotopy equivalence.

MSC:

55P10 Homotopy equivalences in algebraic topology
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