a)
b)
c)
d) no, since R is connected and Q is disconnected
because the continuous image of a connected set must be connected
e)hmmm
I need help with these questions
1. Does there exist a continuous function from (0,1) onto [0,1]
2. Does there exist a continuous function from (-1,1) onto R
3. Does there exist a continuous function from R onto (-1, 1)
4. Does there exist a continuous function from R onto Q
5. Does there exist a continuous function from Q onto R
where R = real numbers
and Q = rational numbers
Of course not. A general theorem (requiring Zorn's lemma) implies that if is surjective then there exists an injection . Does there exist an injection .
And since this thread is labeled topology I suppose that I can impose any topology on these sets.
Give the discrete topology and the indiscrete. Then any mapping is continuous.
I think this may be a confusion. The well ordering principle is usually taken to be: any non-empty subset of the naturals has a first element (in the usual, natural ordering), and this is logically equivalent to the (weak) principle of induction.
Perhaps what Plato assumed, and this is what I thought at first, is Zermelo's well ordering theorem: every non-empty set can be well-ordered , and this in ZF is equivalent to Zorn's lemma and to AC.
Tonio