# Convergence in Topological Spaces .... Singh, Example 4.1.1 .... ....

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In summary, Tej Bahadur Singh discusses the concept of a neighborhood and provides the relevant definitions for readers to have access to. He also provides an example to show that a sequence does not converge to a certain rational number.
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I need help in order to fully understand an example concerning convergence in the space of real numbers with the co-countable topology ...
I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 4, Section 4.1: Sequences ...

I need help in order to fully understand Example 4.1.1 ...Example 4.1.1 reads as follows:

In the above example from Singh we read the following:

" ... ...no rational number is a limit of a sequence in ##\mathbb{R} - \mathbb{Q}## ... ... "My question is as follows:

Why exactly is it the case that no rational number a limit of a sequence in ##\mathbb{R} - \mathbb{Q}## ... ... "
Help will be appreciated ...

Peter=====================================================================================It may help readers of the above post to have access to Singh's definition of a neighborhood and to the start of Chapter 4 (which gives the relevant definitions) ... so I am providing the text as follows:

Hope that helps ...

Peter

Suppose for contradiction that ##\langle x_n\rangle## is a sequence in ##\mathbb{R}\setminus\mathbb{Q}## that converges to a rational number ##x##. Let ##U=\mathbb{R}\setminus\bigcup_{n=1}^\infty \{x_n\}##. This is an open set in ##\mathbb{R}_c## because its complement is the set of the ##x_n##, which is (at most) countable. Also ##x\in U## because ##x## is not any of the ##x_n## (since the ##x_n## are all irrational). So ##U## is an open neighborhood of ##x## that doesn't contain any of the ##x_n##. This means that the ##x_n## do not converge to ##x##.

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Infrared said:
Suppose for contradiction that ##\langle x_n\rangle## is a sequence in ##\mathbb{R}\setminus\mathbb{Q}## that converges to a rational number ##x##. Let ##U=\mathbb{R}\setminus\bigcup_{n=1}^\infty \{x_n\}##. This is an open set in ##\mathbb{R}_c## because its complement is the set of the ##x_n##, which is (at most) countable. Also ##x\in U## because ##x## is not any of the ##x_n## (since the ##x_n## are all irrational). So ##U## is an open neighborhood of ##x## that doesn't contain any of the ##x_n##. This means that the ##x_n## do not converge to ##x##.
Thanks for the help Infrared ...

Peter

## 1. What is convergence in topological spaces?

Convergence in topological spaces refers to the idea that a sequence of points in a topological space can approach a limit point in that space. This means that as the sequence progresses, the points get closer and closer to the limit point, eventually reaching it.

## 2. How is convergence in topological spaces defined?

Convergence in topological spaces is defined using the concept of neighborhoods. A sequence of points converges to a limit point if, for any neighborhood of that limit point, there exists a point in the sequence that is contained in that neighborhood.

## 3. What is the importance of convergence in topological spaces?

Convergence in topological spaces is important because it allows us to study the behavior of sequences in a topological space. It also helps us understand the structure and properties of a topological space.

## 4. Can you give an example of convergence in topological spaces?

One example of convergence in topological spaces is the sequence of rational numbers approaching an irrational number. For example, the sequence 1, 1.4, 1.41, 1.414, ... approaches the limit point √2 in the topological space of real numbers with the standard topology.

## 5. How is convergence in topological spaces related to continuity?

Convergence in topological spaces is closely related to continuity. A function is continuous at a point if and only if it preserves convergence of sequences at that point. This means that if a sequence of points in the domain of a continuous function converges to a limit point, the corresponding sequence of function values in the co-domain also converges to the function value at that limit point.

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