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I'm stuck on an intro to point-set topology homework problem. I'm given that $(X, \mathcal T)$ is a topological space and $p\in X$ has a countable neighborhood basis. I need to then show that $p$ has a nested countable neighborhood basis. The definition of "nested" we are using is: If $\{U_i\}_{i \in \mathbb N}$ is a collection of subsets of a space $X$, we say that $\{U_i\}_{i \in \mathbb N}$ is nested if $U_{i+1} \subset U_i$ for all $i \in \mathbb N$.

Here is my work so far:

$p$ has a countable neighborhood basis $\implies \exists \{ U_\alpha \}_{\alpha \in \lambda}: \lambda \subseteq \mathbb N,$ $p \in U_\alpha \in \mathcal T$ $\forall \alpha \in \lambda$, and $\forall U \in \mathcal T$ with $p \in U$, $\exists \alpha \in \lambda: U_\alpha \subseteq U$. I am now trying to define a nested countable neighborhood basis for $p$ as the sequence of open sets $\{V_n\}_{n \in \mathbb N}$. Any tips on how to do this? Here's what I tried to do:

Define $\{V_n\}_{n \in \mathbb N}$ recursively. First case: $V_1 = U_\alpha$ for some $\alpha \in \lambda$.

Then for $k \geq 1,$ we define $V_{k+1}$. We know that $V_k$ is an open set containing $p$. Let $W$ be an open set containing $p$ such that $W \subset V_k$. Then $W\cap V_k$ is another open set containing $p$, so $\exists \beta \in \lambda: U_\beta \subseteq W \cap V_k \subset V_k$. Define $V_{k+1} = U_\beta$.

We have that $\{V_n\}_{n \in \mathbb N}$ is countable and nested as defined, but is it a neighborhood basis for $p$? We know that $p \in V_n \in \mathcal T$ $\forall n \in \mathbb N$, but do we know that $\forall U \in \mathcal T$ with $p \in U$, $\exists n \in \mathbb N: V_n \subseteq U$? Or maybe I have defined $\{V_n\}_{n \in \mathbb N}$ incorrectly?

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  • $\begingroup$ And, without loss of generality, $\lambda = \mathbb{N}$. That makes the writing simpler. $\endgroup$ – Daniel Fischer Apr 17 '14 at 11:03
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While this approach will mostly work (there is some difficulty in defining $V_{k+1}$ from $V_k$ since you really don't say how the open set $W$ comes into being), you are working much too hard. You are not asked to show that some subcollection of $\{ U_\alpha \}_{\alpha \in \lambda}$ is a countable nested neighbourhood basis; you are only asked to show that one exists, perhaps by constructing a new countable neighbourhood basis (which happens to be nested) from the given one.

I'll leave you with one final hint: the intersection of finitely many open sets is open.

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Actually, if a topological space X has a countable neighborhood base at a point x in X for example, say {U(n)} n in N, then it is easy to construct a collection of neighborhoods with desired properties. We can take V(1)=U(1), V(2)=U(1)∩U(2), V(3)=U(1)∩U(2)∩U(3), ..., V(n)=U(1)∩....∩U(n), and so on.

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