Help with 2 problems about compact/pointwise convergence from Munkres - Topology

For the second problem, if {fn} is equicontinuous, then f is continuous and fn converges to f in the topology of compact convergence.In summary, the conversation discussed two problems from Munkres' book and their convergence in three different topologies: uniform, compact convergence, and pointwise convergence. The first problem was solved for both given sequences, while the second problem was solved by showing that if the sequence is equicontinuous, then the limit function is also continuous and the sequence converges in the topology of compact convergence.
  • #1
Hells_Kitchen
62
0
Hi everyone,

I am stuck with 2 problems from Munkres' book and I would appreciate if someone helped me solve them. Thank you in advance. Here they are:

1. Consider the sequence of continuous functions fn : ℝ -> ℝ defined by fn(x) = x/n . In which of the following three topologies does this sequence converge: uniform, compact convergence, pointwise convergence? Answer the same question for the sequence given as:

fn(x)= 1 / [n^3 * (x - 1/n)^2 + 1]

2. Let (Y,d) be a metric space; let fn: X -> Y be a sequence of continuous functions; let f: X -> Y be a function (not necessarily continuous). Suppose that fn converges to f in the topology of pointwise convergence. Show that if {fn} is equicontinuous then f is continuous and fn converges to f in the topology of compact convergence.
 
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  • #2
Thank you again for your help! For the first problem, 1. fn(x) = x/n converges in pointwise convergence, but not in uniform or compact convergence. 2. fn(x)= 1 / [n^3 * (x- 1/n)^2 + 1] converges in pointwise convergence, but not in uniform or compact convergence.
 

1. What is compact convergence in topology?

Compact convergence in topology refers to a type of convergence where a sequence of functions or elements converges uniformly on a compact set. This means that for any given epsilon, there exists a natural number N such that for all x in the compact set, the distance between the limit of the sequence and the function at x is less than epsilon.

2. How is compact convergence different from pointwise convergence?

Pointwise convergence refers to a type of convergence where a sequence of functions or elements converges at each point individually. This means that for any given x, the distance between the limit of the sequence and the function at x approaches 0 as the sequence progresses. Compact convergence, on the other hand, requires the convergence to occur uniformly across a compact set, rather than at each point individually.

3. What is the significance of compact convergence in topology?

Compact convergence is important in topology because it allows us to prove that a sequence of functions or elements converges to a limit function or element. It also allows us to study the properties of the limit function or element, such as continuity or differentiability, by using the properties of the sequence of functions or elements.

4. What are some examples of compact convergence in topology?

An example of compact convergence is the sequence of functions f_n(x) = x^n on the compact set [0,1]. This sequence converges uniformly to the limit function f(x) = 0 as n approaches infinity. Another example is the sequence of functions f_n(x) = sin(nx) on the compact set [0, 2π]. This sequence also converges uniformly to the limit function f(x) = 0 as n approaches infinity.

5. How can one prove compact convergence in topology?

To prove that a sequence of functions or elements converges compactly, one can use the definition of compact convergence and show that for any given epsilon, there exists a natural number N such that for all x in the compact set, the distance between the limit of the sequence and the function at x is less than epsilon. This can be done by using techniques such as the triangle inequality or bounding the distance between the sequence and the limit function or element.

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