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In the book Topology and Geometry by Bredon, the following is offered as an example of a topological space on page 6 in chapter 1:

$X = \omega \cup \{\omega\}$ with the open sets being all subsets of $\omega$ together with complements of finite sets. (Here, $\omega$ denotes the set of natural numbers.)

I am unsure of the reason for including the nested set $\{\omega\}$ within this definition. Is this standard notation? Does the author truly mean that $X=\{\omega,1,2,3,...\}$? If so, why is the inclusion of $\omega$ important for the construction?

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    $\begingroup$ What they mean is $\omega$, which is $0,1,2,3,4,...$ with that order, together with an extra element $\omega$ which is considered larger than all the others. So, the ordered set $0,1,2,3,4,...,\omega$. Maybe you would like to call that last element $\infty$ in your mind. See, ordinal numbers. Maybe they authors assumed the reader's familiarity with ordinal numbers or at least notation used for them. $\endgroup$
    – user752802
    Mar 8, 2020 at 2:23
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    $\begingroup$ The extra point $\omega$ in the example has the property that all open sets that contain it have finite complement. Unlike the other points $k=0,1,2,3,...$ which are all contained in singletons $\{k\}$ that are open and closed. $\endgroup$
    – user752802
    Mar 8, 2020 at 2:29
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    $\begingroup$ While in your example, the order in $\omega\cup\{\omega\}$ is no relevant in the topology, in example $8$ further ahead it is, where they give it the order topology. $\endgroup$
    – user752802
    Mar 8, 2020 at 2:34
  • $\begingroup$ Thanks! This makes a lot more sense now. $\endgroup$ Mar 8, 2020 at 4:06

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