# Topology: Nested, Compact, Connected Sets

• jjou
In summary, the conversation discusses proving that the intersection of nested, compact, connected sets is nonempty and connected. The first part of the proof uses the assumption that there is a limit point in the sequence (x_n) and shows that this point is in the intersection. The second part uses a contradiction argument to show that the intersection is connected, either by taking the limit of the sequence of intersections or by considering the finite intersection property. The conversation also addresses a typo and clarifies the definition of a limit point.
jjou
[SOLVED] Topology: Nested, Compact, Connected Sets

1. Assumptions: X is a Hausdorff space. {K_n} is a family of nested, compact, nonempty, connected sets. Two parts: Show the intersection of all K_n is nonempty and connected.

That the intersection is nonempty: I modeled my proof after the widely known analysis proof. I took a sequence (x_n) such that $$x_n\in K_n$$ for all n. Assuming x_n has a limit point x (AM I ALLOWED TO ASSUME THE SEQUENCE HAS A LIMIT POINT?), then x is in the sequential closure of K_n, which is contained in the closure of K_n, which is equal to K_n: $$x \in SCl(K_n) \subset Cl(K_n) = K_n$$ (since X is Hausdorff, all compact sets are closed). Thus $$x\in K_n$$ for all n, so it is in the intersection. Therefore the intersection is non-empty. This all hinges on the fact that I assumed there was a limit point ... am I talking in circles, or is this okay?

That the intersection is connected: I'm guessing I should be using contradiction. So, suppose the intersection $$K=\bigcap^{\infty}K_n$$ is not connected, then there exists open sets U, V such that $$U\cap V=\emptyset$$, $$U\cap K\neq\emptyset$$, $$V\cap K\neq\emptyset$$, and $$K\subset U\cup V$$. I also know then that $$U\cap K_n\neq\emptyset$$ for any n and likewise for V. But I don't know that there is any n for which $$K_n\subset U\cup V$$ - which would be the contradiction I am looking for, since every K_n is connected. Or is this not the right method at all?

Last edited:
What do you mean by "limit point"?

For connectedness, we require our set to be in the union of U and V, not their intersection (which is empty!).

Eek, you're right. That was a horrible typo.

By "limit point," I mean any point such that any neighborhood of that point contains points of the sequence... I think.

For showing that the intersection is connected, two ideas - can somebody check them?

METHOD 1
$$K_1 \cap K_2 = K_2$$ is connected. Then, for any $$n\in\mathbb{N}$$, we have $$\bigcap_{i=1}^{n} K_i = K_n$$ is connected. Then let $$n\rightarrow\infty$$..? Or is that oversimplifying the problem?

METHOD 2
Suppose not connected. There exists open sets U, V such that (all assumptions from above). Then consider $$U\cup V$$. There must exist n such that $$K_n\subset U\cup V$$ since {K_n} is a decreasing sequence of nested subsets. In other words, I can view the intersection as the "limit" of the sequence of intersections $$I_n=\bigcap_{i=1}^n K_i$$. Thus, any neighborhood containing K must also contain an element of the sequence. So I take $$U\cup V$$ as my neighborhood containing K and then get my contradiction..?

Please check these for me. Still having trouble with showing K is nonempty. Can someone please offer a hint? Thank you! :)

I got that K is nonempty using the finite intersection property, so part 1 is done.

Still wondering about part 2 (connectedness). Can someone please check the ideas I posted previously?

Nevermind, got it. :)

## What is a nested set in topology?

A nested set in topology is a set that contains another set within it. In other words, it is a set that is a subset of another set. This concept is important in understanding the relationship between different sets in topology.

## What does it mean for a set to be compact in topology?

In topology, a set is considered compact if it is closed and bounded. This means that the set contains all of its limit points and can be contained within a finite distance. Compact sets are important in understanding continuity and convergence in topology.

## What is the definition of a connected set in topology?

In topology, a connected set is a set that cannot be divided into two nonempty subsets that are completely separated from each other. This means that there is no gap or discontinuity between the two subsets. Connected sets are important in understanding the continuity and path-connectedness of a space.

## How does the concept of nested sets relate to compact sets in topology?

The concept of nested sets is closely related to compact sets in topology. This is because a nested set is a subset of another set, and a compact set is a closed subset. Therefore, nested sets are also compact sets. In other words, every nested set is a compact set, but not every compact set is a nested set.

## What are some real-life applications of topology and its concepts of nested, compact, and connected sets?

Topology has various applications in different fields, such as physics, biology, and computer science. For example, in physics, topology is used to study the properties of materials, such as their conductive and insulating properties. In biology, topology is used to understand the shapes and structures of molecules and proteins. In computer science, topology is used in data analysis and image recognition. The concepts of nested, compact, and connected sets are important in understanding the relationships between different data points and structures.

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