Proving Closure of T in B[0,1] Set V

In summary, we are trying to prove that the set T, which is an element of the metric space B[0,1], is closed. B[0,1] is the set of bounded functions from [0,1] to the reals, and the topology is given by the sup norm. The evaluation map taking a function to its value at 1 is continuous, and the inverse image of 2 under this map is closed. We can also try to prove this by taking the complement of the set V=[B[0,1];f(1)=2] and using open or closed balls.
  • #1
Ed Quanta
297
0
Let T be an element of B[0,1] be the set V=[B[0,1];f(1)=2]. Prove that T is closed (in metric space B[0,1]).

I am not sure if it is obvious since I am new to this stuff but B[0,1] is an open ball I believe.

My question is how do I find the complement of V. If I could define B/T then I am hoping it will follow easily from the definitions of open set that this is an open set and my proof will be complete. Is B/T=[B[0,1]; f(1)>2 V f(1)<2]? Very confused.
 
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  • #2
What is B[0,1], the set of functions on [0,1] into R? Are they continuous? What is the metric?
 
  • #3
standard stuff would be something like this:

1) B[0,1] is the set of continuous functions from [0,1] to the reals

or maybe because there is a B there, it is just bounded functions.

2) the topology on B[0,1] is given by say the sup norm, i.e. the distance from f to g is the furthest apart any two values f(t), g(t) ever get for t in [0,1].
3) then prove the evaluation map taking f to f(1) is continuous.

4) then the inverse image of 2 under the evaluation map is clsoed since 2 is closed in R.
 
  • #4
Thanks a lot mathwonk
 
  • #5
What if I were to try to solve this by taking the complement of V and using open or closed balls? Is there any way to prove it this way? B[0,1] is the set of bounded functions by the way.
 

Related to Proving Closure of T in B[0,1] Set V

1. How do you define "closure" in this context?

Closure in this context refers to the smallest closed set that contains all the elements of a given set V, with respect to a given topology T. In other words, it is the set of all limit points of V under T.

2. Why is proving closure important in mathematics and science?

Proving closure is important because it allows us to determine the completeness and compactness of a set, which are key properties in mathematical analysis. It also helps to establish the existence and uniqueness of solutions in various scientific fields, such as physics and engineering.

3. What is the process for proving closure of a set in a given topology?

The process for proving closure of a set V in a given topology T involves showing that every limit point of V is contained in the closure of V, and every element in the closure of V is either a limit point or an element of V. This can be done by constructing a sequence of points in V that converges to a limit point in the closure of V.

4. How does proving closure of a set relate to continuity?

Proving closure of a set is closely related to continuity, as it helps to establish the continuity of a function or mapping between two topological spaces. In particular, a function is continuous if and only if its pre-image preserves closure, meaning that the closure of the pre-image is contained in the pre-image of the closure.

5. Can you provide an example of proving closure in a real-world scenario?

One example of proving closure in a real-world scenario is in the construction of a bridge. The engineer must ensure that the set of all possible loads on the bridge is closed, meaning that all possible load combinations can be supported by the bridge without causing it to collapse. This requires proving closure of the set of all possible loads under a given topology, such as the limit states design method.

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