# Meaning of notation for intersection of set with a nested version of itself

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?

• 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.
– user752802
Mar 8, 2020 at 2:23
• 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.
– user752802
Mar 8, 2020 at 2:29
• 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.
– user752802
Mar 8, 2020 at 2:34
• Thanks! This makes a lot more sense now. Mar 8, 2020 at 4:06