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Mathematics
Topology and Analysis
Convergence .... Singh, Example 4.1.1 .... .... Another Question ....
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[QUOTE="Math Amateur, post: 6312840, member: 203675"] [B]TL;DR Summary:[/B] 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: [ATTACH type="full"]258921[/ATTACH]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: [ATTACH type="full"]258922[/ATTACH] [ATTACH type="full"]258923[/ATTACH] [ATTACH type="full"]258924[/ATTACH] Hope that helps ... Peter [/QUOTE]
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Mathematics
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Convergence .... Singh, Example 4.1.1 .... .... Another Question ....
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