Open set of real valued functions

In summary, the question asks to prove that the set A of all continuous functions on [0,1] with a distance metric defined as the supremum of the absolute value of the difference between the functions is open. To prove this, it is necessary to show that every point in A is an interior point, which means that there exists a neighborhood of that point that is entirely contained within A. The set A is defined as the subset of X where the function evaluated at 0 is greater than 1. Therefore, to find a neighborhood of a function in A, one would need to find other functions in X that are also in A and have a distance from the original function that is less than the distance between the function and the boundary
  • #1
rjw5002

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.
 
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  • #2
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?
 

FAQ: Open set of real valued functions

What is an open set of real valued functions?

An open set of real valued functions is a set of functions where every function in the set can be continuously varied within a certain range without reaching the boundary of the set.

How is an open set of real valued functions different from a closed set?

An open set of real valued functions does not include its boundary points, while a closed set includes all of its boundary points.

What is the importance of open sets in the study of real valued functions?

Open sets are important in the study of real valued functions because they provide a way to define and analyze continuous functions, which are fundamental in many areas of mathematics and science.

How are open sets related to limits and continuity of real valued functions?

Open sets are closely related to the concepts of limits and continuity in real valued functions. A function is continuous at a point if and only if the preimage of every open set containing the output of that point is an open set in the domain of the function.

Are there any practical applications of open sets in real world problems?

Open sets have practical applications in various fields such as physics, engineering, economics, and computer science. For example, in physics and engineering, open sets are used to describe the behavior of continuous systems. In economics, open sets are used to model the behavior of markets and prices. In computer science, open sets are used in the analysis of algorithms and data structures.

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