what is the definition of local base

Question

Hi guys I am facing problem while understanding the definition of a local base. Can anybody tell me the difference/relation between the following two definitions? Def $1$: "Let $p$ be any arbitrary point in a topological space $X$. Then a class $B_p$ of OPEN SUBSETS of $X$ is called a local base at $p$ iff for each open set $U$ containing $p$ there exists $G_p$ in $B_p$ such that $p$ $\in$ $G_p$ $\subseteq$ $U$. The second definition which I found in other books includes NEIGHBOURHOODS OF $P$ instead of open sets.

Answer 1

If we have an "open local base", it is of course a "neighbourhood local base" as well.

This is because $N$ is a neighbourhood of $x \in N$ iff there exists an open set $O$ such that $x \in O \subseteq N$. So by definition (take the set itself every time), an open set is a neighbourhood for each of its points (and this is indeed the definition of open, if you define a topology via so-called neighbourhood systems). For given $N$, the subset of its points that $N$ is a neighbourhood of, is called $\operatorname{int}(N)$, the interior of $N$, which is an open set, often also defined as $\operatorname{int}(N) = \cup\{O: O \text{ open, } O \subseteq N\}$, which makes its openness apparent.

On the other hand, if we have a "neighbourhood local base" $\mathcal{N}$ for $x$, the set $\mathcal{B} = \{\operatorname{int}(N): N \in \mathcal{N}\}$ is an "open local base" for $x$: if we have any open set $U$ containing $x$, we have $x \in N \subseteq O$ for some $N \in \mathcal{N}$, and then also $x \in \operatorname{int}(N) \subseteq N \subseteq U$ as well, as $N$ is a neighbourhood of $x$ by assumption.

So one is a restricted version of the other (demanding the sets to be open is slightly more restrictive than mere neighbourhoods), but the more liberal version can be handy in some more compact formulations, e.g.:

a Hausdorff space $X$ is locally compact iff it has a local base of compact neighbourhoods.

A space is regular iff every point has a local base of closed neighbourhoods.

But as you saw, a "neighbourhood local base" can be made into an open one, taking interiors , so properties like being first countable are not affected by the definition (if you're first countable in one form then also in the other).