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I'm a little bit confused about the point of local base for which definition defined as in here and the answer here. I made some example and came to the below statement and example.

Statement: While neighborhoods are a general concept that exists for every point, a local base is a more specific and structured collection of open sets designed to describe the local topology around a particular point.

Example:

Consider the topological space X = {a, b, c} with the topology τ={∅,{a},{b},{a,b},{a,b,c}}. This is a simple space where {a,b} is the only non-trivial open set.

Point a:

Neighborhoods: Every open set containing a is a neighborhood of a. Therefore, {a}, {a,b}, and {a,b,c} are neighborhoods of a.

Local Base: In this case, we can define a local base for a explicitly as {{a},{a,b}}. This collection satisfies the conditions of a local base at a.

Point c:

Neighborhoods: Every open set containing c is a neighborhood of c. Therefore, {c} and {a,b,c} are neighborhoods of c.

Local Base: It's not possible to find a local base for c in this topology. A local base should contain open sets that are nested within each other, covering c and fitting into any larger open set containing c. However, in this example, there are no open sets smaller than {c} that could form part of a local base for c.

The question is whether the above statement and example correct? Thanks in advance.

Bohdan_
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  • I'd only call the empty set and the whole space trivial open sets. Insofar, ${a}$ and ${b}$ are also non-trivial. In fact, every neighbourhood of $a$ contains the open neighbourhood ${a}$ as a subset, hence ${{a}}$ is also a local base. – Hagen von Eitzen Nov 13 '23 at 16:11

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No, this is not right. What is the point of a local base? points out that, for example, the set of all open sets containing $x$ is a local base for $x$. So any point in any topology has one. The point is that some are more convenient than others.

Your local base for $a$ is fine.

A local base for $c$ would be $\{\{a, b, c\}\}$ (in fact that's the only one). $\{c\}$ is not open in your topology, so it's not a neighborhood of $c$.

Also, note that although each element $N$ of a local base must be a neighborhood, $N$ does not have to be open; it may be larger than some open set.

Hew Wolff
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