Property Suggestion
A space $(X, \tau)$ is said to have finitely many open sets if its topology is finite, i.e. $|\tau|<\infty$.
Rationale
I was looking for examples of spaces that are both Noetherian and Artinian as they seemed like dual properties. The current spaces that pi-base has that are both Noetherian and Artinian are either finite or indiscrete (Explore). I initially thought that we can turn this into a theorem, however there's a simple counterexample: take $X$ to be any infinite set, $p\in X$ a distinguished point and let $\tau = \{\emptyset, \{p\}, X\}$. Ultimately, being Noetherian or Artinian can be reduced to a property of open sets rather than the cardinality of the space. It's probably worth adding such a space too if we add this property.
Relationship to other properties
We certainly have that Finite => Has finitely many open sets. We can replace a bunch of theorems that assume finite with those that just assume Has finitely many open sets.
- Has finitely many open sets => Artinian
- Has finitely many open sets => Noetherian
- Has finitely many open sets => Compact
- Has finitely many open sets => Second countable
- etc,
We also have:
- Indiscrete => Has finitely many open sets
- Artinian $\wedge$ Noetherian => Has finitely many open sets
- Has finitely many open sets $\wedge$ ~Finite => ~Door
- Has finitely many open sets $\wedge$ T0 => Finite
- certainly a bunch more
For pretty much all spaces we can easily conclude whether we have this property or not. This property is probably not high priority, though.
Property Suggestion
A space$(X, \tau)$ is said to have finitely many open sets if its topology is finite, i.e. $|\tau|<\infty$ .
Rationale
I was looking for examples of spaces that are both Noetherian and Artinian as they seemed like dual properties. The current spaces that pi-base has that are both Noetherian and Artinian are either finite or indiscrete (Explore). I initially thought that we can turn this into a theorem, however there's a simple counterexample: take$X$ to be any infinite set, $p\in X$ a distinguished point and let $\tau = \{\emptyset, \{p\}, X\}$ . Ultimately, being Noetherian or Artinian can be reduced to a property of open sets rather than the cardinality of the space. It's probably worth adding such a space too if we add this property.
Relationship to other properties
We certainly have that Finite => Has finitely many open sets. We can replace a bunch of theorems that assume finite with those that just assume Has finitely many open sets.
We also have:
For pretty much all spaces we can easily conclude whether we have this property or not. This property is probably not high priority, though.