Re: Choice question; Was: metric spaces and convergent sequences

Stephen J. Herschkorn wrote:

Does the proof of the following require the Axiom of Choice?

Let A be subset of a topological space. If A contains
all limits of all convergent nets in A, then A is closed.

A good reference for these kinds of questions is:

Horst Herrlich, "Axiom of Choice", Lecture Notes in
Mathematics #1876, Springer-Verlag, 2006, xiv + 194 pages.
http://www.amazon.com/dp/3540309896

----------------------------------------------------

The following ZF-without-choice possibilities appear
on p. 73 of Herrlich's book:

1. R can fail to be Frechet, i.e. not every accumulation point x
of a subset A may be reachable by a sequence in A.

2. R may fail to be sequential, i.e., there may be non-closed,
sequentially closed subsets of R. [A set is sequentially
closed means that no point outside of the set can be the
limit of a sequence of points in the set.]

3. R may fail to be Lindelof.

4. All Lindelof subspaces of R may be compact.

5. Subspaces of R may fail to be separable.

6. Complete subspaces of R may fail to be closed in R.

7. Sequentially compact subspaces of R may fail to be bounded
or to be closed in R.

8. Functions f:R --> R may be sequentially continuous at
some point x, but fail to be [epsilon-delta] continuous
at x.

9. Functions f:X --> R, defined on some subspace X of R,
may be sequentially continuous, but fail to be continuous.
[I'm not sure what X is allowed to be. From #8 below, X
can't be R, so I would think X can't be any open interval.]

10. Infinite subsets of R may be D-finite.

----------------------------------------------------

The following ZF theorems appear on p. 72 of Herrlich's book:

1. R and all its subspaces are metrizable, hence normal.

2. R and all its subspaces are second countable.

3. R is separable.

4. A subspace of R is connected if and only if it is
an interval.

5. A subspace of R is compact if and only if it is closed
and bounded in R.

6. Each bounded infinite subset of R has an accumulation
point.

7. R is a countable union of compact sets.

8. A function f:R --> R is continuous if and only if
it is sequentially continuous.

9. A function f:[0,1] --> R is continuous if and only if
it is uniformly continuous.

----------------------------------------------------

Dave L. Renfro
.

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