I'm working on the exercises of Munkres' Topology. Section 30, exercise 1-a says: "Show that in a first-countable space, every one-point set is a set"
Now, I've seen the solution to this exercise and I believe it's wrong, because it assumes that a countable basis at x is a family of sets that get "arbitrarily small", that is, it's like a countable collection of balls of radius 1/n (I know the space doesn't need to be metrizable, I'm just giving the intuition). This assumption is wrong because there could be a strange topology in in which the neighbourhoods of 0 would be intervals of the form (-1-1/n,1+1/n). This is a countable basis at 0.
Now the counter example: with the finite complement topology. In this topology, it is and first countable, but no one-point set is a set.
Is this counter-example right? Thanks in advance
I will transcribe the definition from the book.
A space X is said to have a countable basis at x if there is a countable collection B of neighborhoods of x such that each neighborhood of x contains at least one of the elements of B. A space that has a countable basis at each of its points is said to satisfy the first countability axiom, or to be first-countable.
I think that the family I defined is a countable family of neighborhoods of x, such that each neighborhood contains at least one other (in fact infinite neighborhoods). Since x is arbitrary, that would make the space first-countable.
PS: Neighborhood is defined just as an open set containing x, and not a set which contains an open set containing x.
I will simply prove that with the cofinite topology cannot be first countable. Suppose that and is a local basis at
Thus must be countable.
So must be uncountable and contain a point such .
But is an open set and .
Now it is impossible for .
I'm not sure if you read my first post without attention or if I wasn't clear, but my contention is that a countable basis at x is not a collection of sets that gets arbitrarily small. That's why it's difficult to do the exercise.
Help would be apreciated. No time constraints, this isn't a homework assignment.