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Suppose that $X$ is a space filtered by closed subspaces $X_{1}\subset X_{2}\subset \dots$.

As topological space $X=\operatorname{colim}_{n}X_{n}$. We define $Y_{n}=X_{n+1}/X_{n}$, and consider the induced maps $Y_{n}\rightarrow Y_{n+1}$. Let define $Y=\operatorname{colim}_{n}Y_{n}$.

Question: $X$ is homeomorphic to $Y$?

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  • $\begingroup$ What does the quotient notation $X_{n + 1}/X_n$ mean? $\endgroup$ – LSpice May 30 '19 at 0:11
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As far as I can tell, this is not true in general.

Consider simply $X_n=X$. In this case $Y_n$ (and consequently $Y$) will be a singleton.

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  • $\begingroup$ You are right! My question question is stupid. $\endgroup$ – TTip May 27 '19 at 15:26

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