Closure and connected spaces

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In summary, closure in mathematics refers to the smallest closed set that contains all the elements of the original set. It is closely related to connected spaces, as a set is considered connected if its closure is also connected. The closure of a set can vary depending on the topology used to define it, and it can be calculated by taking the union of the set and its limit points. Understanding closure and connected spaces is crucial in many areas of mathematics, including topology, analysis, and geometry.
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If the closure of a space C is connected, is C connected?
 
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What do you think?? Can you come up with a counterexample?? (this should suggest that the answer is no)
 
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What is the usual way to show that the circle is not homeomorphic to the real line?
 
  • #4
Can you think of some dense subsets of the real line?
 
  • #5
blahblah8724 said:
If the closure of a space C is connected, is C connected?

There's a really easy counterexample. Just take the real line and ______ one single _______.
 

1. What is meant by "closure" in mathematics?

The closure of a set in mathematics is the smallest closed set that contains all the elements of the original set. In simpler terms, it is the set of all points that can be reached from the original set by continuous transformations.

2. How is closure related to connected spaces?

Closure and connected spaces are closely related concepts. In fact, a set is considered connected if and only if its closure is also connected. This means that a set is connected if it cannot be divided into two disjoint open sets.

3. Can a set have a different closure in different topologies?

Yes, a set can have a different closure in different topologies. This is because the closure of a set is dependent on the topology used to define it. Different topologies can lead to different sets of points being included in the closure of a given set.

4. How is the closure of a set calculated?

The closure of a set can be calculated by taking the union of the set and its limit points. Limit points are points that can be approximated by points in the set. In other words, they are points that are very close to the set but may not be contained in it.

5. What is the importance of closure and connected spaces in mathematics?

Closure and connected spaces are important concepts in mathematics as they help us understand the structure and behavior of sets. They also have many applications in different fields of mathematics, such as topology, analysis, and geometry.

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