Finite complement topology
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Definition
Let [ilmath]X[/ilmath] be an arbitrary set. There is a topology, [ilmath]\mathcal{J} [/ilmath], we can give [ilmath]X[/ilmath] called "the finite complement topology", such that [ilmath](X,\mathcal{J})[/ilmath] is a topological space. It is defined as follows[1]:
- [ilmath]\mathcal{J}:\eq\left\{U\in\mathcal{P}(X)\ \big\vert\ U\eq\emptyset \vee\vert X-U\vert\in\mathbb{N}\right\} [/ilmath][Note 1][Note 2], that is to say [ilmath]U\in\mathcal{P}(X)[/ilmath] is in [ilmath]\mathcal{J} [/ilmath] if [ilmath]U\eq\emptyset[/ilmath] or the complement of [ilmath]U[/ilmath] in [ilmath]X[/ilmath] has finite cardinality.
Hence the name "finite complement topology"
A topology must contain the empty set. Hence the first condition, note that [ilmath]X-\emptyset\eq X[/ilmath] which may not be finite! Thus [ilmath]\emptyset[/ilmath] might not otherwise be there.
See also
Notes
- ↑ Many authors give the [ilmath]U\eq\emptyset[/ilmath] condition as [ilmath]X-U\eq X[/ilmath]. It is easy to see however that:
- [ilmath][X-U\eq X]\iff[U\eq\emptyset][/ilmath]
- ↑ We write [ilmath]X-U[/ilmath] for set complement of [ilmath]U[/ilmath] in [ilmath]X[/ilmath]. Rather than [ilmath]U^\mathrm{C} [/ilmath] or something. This helps with subspaces.
