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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

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