verify the assertion. (Subspace example)
1) The set of continuous real-valued functions on the interval
[0,1] is a subspace of R^[0,1]
This is from Linear Algebra Done Right- Sheldon Axler 3rd
edition.
I don't understand why the solution uses a integral.
3) Prove that the cardinality of the open unit interval, (0,1),
is equal to the cardinality of the open unit cube:
{(x,y,z) E R^3|0<x<1, 0<y<1,
0<Z<1}.
[Hint: Model your argument on Cantor's proof for the
interval and the open square. Consider the decimal expansion of the
fraction 12/999. It may prove handdy]
Prove or disprove that the union of two subspaces is a subspace.
If it is not true, what is the smallest subspace containing the
union of the two subspaces.
Topology
Prove or disprove ( with a counterexample)
(a) The continuous image of a Hausdorff space is Hausdorff.
(b) The continuous image of a connected space is
connected.
Consider R with the cofinite topology. Is [0,1] compact? Can you
describe the compact sets?
Consider R with the cocountable topology. Is [0, 1] compact? Can
you describe the compact sets?
Consider R with the lower limit topology. Is [0,1] compact? Can
you describe the compact sets?