# Convergence .... Singh, Example 4.1.1 .... .... Another Question ....

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In summary, the conversation is about understanding Example 4.1.1 in Chapter 4, Section 4.1: Sequences from Tej Bahadur Singh's book "Elements of Topology". The conversation includes a question about the complement of a neighborhood and the existence of an integer that satisfies a certain condition. The response includes a summary of the relevant definitions and an additional explanation about the concept of convergence in topological spaces.
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I need further 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 some further 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:

" ... ...Then the complement of ##\{ x_n \ | \ x_n \neq x \text{ and } n = 1,2, ... \}## is a nbd of ##x##. Accordingly, there exists an integer ##n_0## such that ##x_n = x## for all ##n \geq n_0##. ... ... "My question is as follows: Why, if the complement of ##\{ x_n \ | \ x_n \neq x## and ##n = 1,2, ... \}## is a nbd of ##x## does there exist an integer ##n_0## such that ##x_n = x## for all ##n \geq n_0##. ... ... ?Help will be much 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

##V=\mathbb{R} \setminus \{x_n: x \neq x_n, n \geq 1\}## is a neighborhood of ##x## as ##V## contains ##x## and it is the complement of a countable set, thus open.

But ##x_n \to x##, which means that for all neighborhoods ##U## of ##X##, there is ##n_0## such that ##x_n \in U## if ##n \geq n_0##.

Next, take ##U = V##. Then there is ##n_0## such that ##x_n \in V \iff x_n = x## if ##n \geq n_0##.

__________________

Note that the converse of what the author claims holds: any sequence that is eventually constant converges (to the eventually constant value). Thus in this topological space, the convergent sequences are exactly these which are eventually constant.

On ##\mathbb{R}##, we can also define the discrete topology ##\mathcal{P}(\mathbb{R})## (every subset of ##\mathbb{R}## is open). Then also a sequence in ##\mathbb{R}## is convergent if and only if it is eventually constant (easy exercise).

Thus we see that ##(\mathbb{R}, \mathcal{T}_c)## and ##(\mathbb{R}, \mathcal{P}(\mathbb{R}))## are different topological spaces with the same convergent sequences.

This means that sequences do not suffice to describe the topology (unlike for metric spaces or more generally spaces where every point has a countable neighborhood basis). The solution then is to generalise the concept of sequence and allow more general index sets than ##\mathbb{N}##. We then come to the concept of nets and these do describe the topology. But some authors use other approaches and use filters instead of net, though both concepts are equivalent and can be translated to one another.

TLDR: Sequences in topological spaces do not suffice to describe the topology, so there is the need to introduce a new kind of object to describe convergence. This will be a net or a filter.

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Thanks for the help, Math_QED ...

And thanks also for a most interesting post ...

Peter

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## 1. What is convergence in the context of science?

Convergence in science refers to the merging of different fields and disciplines to address a common problem or advance a particular area of research. It involves collaboration and integration of knowledge, methods, and technologies from multiple disciplines to achieve a greater understanding of complex systems.

## 2. Why is convergence important in scientific research?

Convergence allows scientists to approach problems and challenges from multiple perspectives, leading to more comprehensive and innovative solutions. It also promotes interdisciplinary collaborations, which can lead to breakthrough discoveries and advancements in various fields.

## 3. Can you provide an example of convergence in science?

One example of convergence in science is the use of nanotechnology, biology, and engineering to develop targeted drug delivery systems. This approach combines knowledge and techniques from these different fields to create more effective and efficient treatments for diseases.

## 4. How does convergence benefit society?

Convergence has the potential to benefit society in various ways, such as improving healthcare, developing sustainable technologies, and addressing global challenges. It can also lead to the creation of new industries and job opportunities.

## 5. What are the challenges of convergence in science?

Some challenges of convergence in science include the need for effective communication and collaboration between different disciplines, securing funding for interdisciplinary research, and navigating ethical considerations. It also requires a shift in traditional academic structures and incentives to support interdisciplinary work.

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