A collection 
- Show that there is an independent family of the same size as the reals. (This means that there is an independent family
. If it is easier to think of it this way, you can use that
of infinite sequences of zeros and ones.)
- Given a set
recall that its characteristic function is the map
such that
if
and
if
We can think of each
as a vector in the space
Suppose now that
is a linearly independent set. (Thus any basis for
over
- Consider
and show that any basis must have the same size as the reals.
A kind of library 
Here is a `hint’ for the first problem: If
is countable, it clearly suffices to find an independent family as required, but consisting of subsets of 
(so we replace each 
with 
of course).
Take
is finite, and 
Check that 
is countable.
Now, given
let 
The claim is that 
is as wanted.
Check that the assignment
is injective.
To show that
satisfies the stronger independence requirement, suppose 
are infinite, pairwise distinct, subsets of 
Write 
For each 
and 
let 
Show that for any finite 
there is an 
such that 
Hmm… Here is an easier way of arguing about the second problem (easier meaning that we do not need problem 1):
An almost disjoint family is a collection of subsets of
that are infinite, but the intersection of any two of them is finite. It is much easier (I think) to come up with an almost disjoint family of the same size as 
than it is to come up with an independent family.
For example: For each real
fix a sequence of (pairwise distinct) rationals converging to 
and let 
be the range of this sequence. Then 
is almost disjoint. (And, of course, 
is countable.)
To solve problem 2, one could start with an almost disjoint (rather than independent)
For problem 3: a. If you assume the Continuum Hypothesis
and answer the problem under this additional assumption, you get some partial credit, but to get full credit you need to answer the question independently of whether 
holds.
b. In case you need this, you may use that all bases have the same size. Again, one can solve the problem without appealing to this fact, but feel free to use it if you need it.
c. Also, in case you need it, feel free to assume the Schröder-Bernstein theorem: If
and 
are sets and we have injections from 
into 
and from 
into 
then 
and 
have the same size (i.e., there is a bijection between the two of them). Again, the problem can be solved without using this fact, but feel free to use it if you see the need.
[…] dimensional linear algebra, that is usually studied in the setting of functional analysis). I once asked for the details of the sketch above on a homework […]