- #1

judelaw

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Any help would be greatly appreciated. Thanks!

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In summary, the countable closed topology is the topology where as closed sets are all countable subsets of X. This is proved to be a topology on X by showing that any subset of X is countable.

- #1

judelaw

- 3

- 0

Any help would be greatly appreciated. Thanks!

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- #2

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What have you already tried to solve this? For being a topology, you need to satisfy three axioms, which ones? And which ones are troubling you?

- #3

judelaw

- 3

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I'm not really sure what countable subsets of X means. I sort of understand the definition of countable (exists a bijection between it and the set of natural numbers?), but I don't really know how to relate that to this problem.

- #4

disregardthat

Science Advisor

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1) Is the empty set an open set? Is X an open set?

2) Is a union of open sets an open set?

3) Is a finite intersection of open sets an open set?

or equivalently:

1) is the empty set and X closed sets?

2) Is a finite union of closed sets a closed set?

4) Is an intersection of closed sets a closed set?

Being countable means to be either finite or in bijection with the natural numbers. Any subset of a countable set is countable.

- #5

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Maybe some examples will help you: in [itex]\mathbb{R}[/itex], we have the following countable subsets: [itex]\mathbb{N}[/itex], [itex]\mathbb{Q}[/itex], {1,3,5,7,9,11,...}. While [0,1] or [itex]\mathbb{R}^+[/itex] are not countable.

So, what are the axioms that the closed sets must satisfy? Well, the first one is that the empty set and X are both closed. Can you show this?

- #6

judelaw

- 3

- 0

Thank you micromass and disregardthat!

And I just found a new website to frequent.

A countable closed topology is a type of topological space in which the number of open sets is countably infinite and every closed set is also open. This means that the space has a countable basis, which is a collection of open sets that can be used to describe the entire topology of the space.

The properties of a countable closed topology include being T1, meaning that every point has a neighborhood that does not contain any other points, and being compact, meaning that every open cover has a finite subcover. It also has the Hausdorff property, meaning that any two distinct points have disjoint neighborhoods.

A countable closed topology is different from other topologies in that it has a countable basis, whereas other topologies may have uncountably infinite bases. It also has the property that all closed sets are also open, which is not true for other topologies.

Some examples of spaces with a countable closed topology include the natural numbers with the discrete topology, the real numbers with the Euclidean topology, and any finite or countably infinite set with the indiscrete topology.

Countable closed topology has applications in many areas of science, including computer science, physics, and biology. It can be used to model the behavior of chaotic systems, analyze the convergence of algorithms, and study the topology of genetic networks. Additionally, countable closed topologies are used in the construction of fractals and in topological data analysis.

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