Open set of real valued functions

  • Thread starter rjw5002
  • Start date
  • #1

Homework Statement


Consider the set X = {f:[0,1] [tex]\rightarrow[/tex] R | f [tex]\in[/tex] C[0,1]} w/ metric d(f,g) = sup|f(x) - g(x)| (x [tex]\in[/tex] [0,1])
Prove that the set A = {f [tex]\in[/tex] X | f(0) > 1} is open in (X,d).



Homework Equations


E is open if every point of E is an interior point
p is an interior point of E if there is a neighborhood N of p s.t. N[tex]\subset[/tex]E.
C[0,1] is the set of all real valued functions on [0,1]
supremum.

The Attempt at a Solution



To be honest, I have a great deal of difficulty understanding this question. Any hints or advice to help me in the right direction would be greatly appreciated.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
43,021
970
If f is a function in A what does a neigborhood of A look like?

If A is the set of all functions, f, such that f(0)> 1, what are the interior points of A?
 

Suggested for: Open set of real valued functions

Replies
17
Views
945
Replies
1
Views
605
Replies
12
Views
236
Replies
5
Views
343
Replies
14
Views
1K
Replies
9
Views
788
Replies
8
Views
911
Replies
5
Views
758
  • Last Post
Replies
17
Views
420
Replies
15
Views
467
Top