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- Jan 26, 2012

- 995

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**Problem**: Consider the following:

Show that the above result isSuppose $X=A_1 \cup A_2 \cup \ldots ,$ where $A_n \subseteq \text{ Interior of } A_{n+1}$ for each $n$. If $f:X \rightarrow Y$ is a function such that $f|A_n:A_n \rightarrow Y$ is continuous with respect to the induced topology on $A_n$, show that $f$ itself is continuous.

**false**if the words "Interior of" are removed. Specifically, give an example of a topological space $X$, an increasing nest of subspaces $A_n$ with $X=A_1 \cup A_2 \cup \ldots ,$ and a function $f:X \rightarrow Y$ such that, for each $n$, $f|A_n:A_n \rightarrow Y$ is continuous with respect to the induced topology on $A_n$, but such that $f$ is

__not__continuous on $X$.

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