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February 3

Definition of a topological space by neighborhoods

In the article on "topological space" the section Definition by Neighborhoods lists four axioms. I don't see how axiom 4 is independent.

Axiom 1 says: If N is a neighborhood of point x, then x is an element of N.

Axiom 4 says: If N is a neighborhood of point x, then N contains a neighborhood M of x such that N is a neighborhood of every point in M.

Doesn't 4 follow directly from 1? The singleton set containing only the point x is a subset of every neighborhood of x, so it always satisfies the conditions for neighborhood M in 4.

What am I missing here? Can you point me to a description of axiom 4 that I can understand without a deep background in topology? — Preceding unsigned comment added by Bronco creek (talk • contribs) 19:01, 3 February 2022 (UTC)[reply]

What you're missing is that the singleton {x} might not be a neighborhood of x. --Trovatore (talk) 19:09, 3 February 2022 (UTC)[reply]

Explicit example: take the Sierpiński space with points {a, b}, open sets ∅, {a}, {a, b}. Then {b} is not a neighbourhood of b. Double sharp (talk) 15:41, 9 February 2022 (UTC)[reply]

Thanks. That'll teach me to trust my intuition. — Preceding unsigned comment added by Bronco creek (talk • contribs) 19:54, 3 February 2022 (UTC)[reply]

The key is not so much not to trust intuition, as to get a more trustworthy intuition.

Intuition is fundamental to mathematics. You simply can't do anything nontrivial without it. But you have to develop it. And you have to keep testing it, looking for places where it doesn't serve you well, and be willing to change those.

In this particular case, the intuition that might have served you is that a neighborhood is a set that contains a given point, plus a little margin around it. That ordinarily rules out singletons. Note that it doesn't always rule out singletons, but that's the base case you can have in mind, being aware of exceptions. --Trovatore (talk) 20:19, 3 February 2022 (UTC)[reply]

The key is not so much not to trust intuition, as to get a more trustworthy intuition. Words to live by. --RDBury (talk) 22:58, 3 February 2022 (UTC)[reply]