Set A={f element C([0,1]): f(x)>0 for all x in [0,1]}

Find the interior and closure of A in C([0,1])

Explain

Please help

Printable View

- Apr 25th 2010, 12:10 PMderek walcottinterior/ closure
Set A={f element C([0,1]): f(x)>0 for all x in [0,1]}

Find the interior and closure of A in C([0,1])

Explain

Please help - Apr 25th 2010, 12:24 PMderek walcott
i am looking at the definitions for interior and kind of confused because it says that in order for S to be interior, Int(S) is an open set of the metric space -X

looking at my metric space, C([0,1]) and my S, A, ...... it looks as thought that can't be true because C is a closed set

grrrrrr - Apr 25th 2010, 12:29 PMAmer
- Apr 25th 2010, 12:55 PMderek walcott
so is.... C([0,1]) a line segment on a closed interval then?

and A is any point in C?

i can't visualize it - Apr 25th 2010, 01:23 PMderek walcott
Int(A) = C((0,1))

and closure of A = C([0,1])

close... correct .... not close at all .....? - Apr 25th 2010, 01:27 PMAmer
what is the whole question I think there is something missing,

what is the base for the space we have.

I solved it but I'm not sure ...

if we say A is the subset of the space that contains all continuous functions at [0,1]

and the functions of A is continuous and has a positive range

A is open in that space since all elements of A is continuous functions at [0,1]

so Int(A) =A

Closure of A = A if A is closed in the space

A is closed that means the complement of A is open

the complement of A is the continuous functions defined at [0,1]

with range equal 0 and negative range

as you can see the complement is open since it contains continuous functions defined at [0,1]

so Closure(A) = A since A is closed to

... - Apr 25th 2010, 01:33 PMderek walcott
professor never said anything about the base of the space

- Apr 25th 2010, 01:35 PMAmer
ok what is the metric defined at the space ??

- Apr 25th 2010, 01:54 PMPlato
Let’s get the notation straight. Here is the usual understanding for this material.

$\displaystyle \mathcal{C}([0,1])$ is the set of continuous functions on $\displaystyle [0,1]$.

The metric is for $\displaystyle \{f,g\}\subset\mathcal{C}([0,1])$ then $\displaystyle d(f,g) = {\max }_{x \in [0,1]}\left| {f(x) - g(x)} \right|$.

P.S.

If $\displaystyle f\in \mathcal{C}([0,1])\cap A$ the because of the properties of continuous functions we know that

$\displaystyle f$ has a**positive minimum**on $\displaystyle [0,1]$. Lets call it $\displaystyle f(t)$.

Now define $\displaystyle \varepsilon = \frac{{f(t)}}{2} > 0$.

Suppose $\displaystyle h\in\mathcal{B}(f; \varepsilon )=\{g:d(f,g)< \varepsilon \}$

Can you show that $\displaystyle h\in A?$