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| ``Sequences in Quotient Spaces''
by t2space on 2003-08-30 05:23:05 |
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| Let $X$ be a topological space and suppose that $\theta$ is an equivalence relation on $X$. Let $p:X\to X/\theta$ denote the
canonical projection. Remember that the quotient topology is the
finest topology on $X/\theta$ such that $p$ is continuous.
Now if $(x_n)_{n\in \omega}$ is a sequence in $X$ converging to
some $x\in X$ then the sequence $(p(x_n))$ converges to $p(x)$. Now
the converse situation seems to be way more difficult.
Let $(\xi_n)$ be a sequence in $X/\theta$ that converges to some $\xi \in X/\theta$. Question: when is there a sequence $(x_n)_{n\in \omega}$ in $X$ converging to some $x\in X$ such that
- $p(x_n) = \xi_n for all n$
- $p(x)= \xi$ ? |
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