# Filtered colimit of a topological space

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 $$Y$$?

• What does the quotient notation $X_{n + 1}/X_n$ mean? – LSpice May 30 '19 at 0:11

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