Generic spaces

[jamesdabbs]

From jamesdabbs/pi-base#43

There are several "generic" topologies in Counterexamples whose properties depend on the underlying set. The entries in the General Reference Chart of Counterexamples are all blank, but in Pi-Base they seem to have been filled in with some unwritten assumptions on the underlying set. Some traits depend only on the definition of openness and not at all on the underlying set, but I suspect these are few. To handle this, we either need to have separate versions with explicit underlying sets, or else an option of "possibly" in addition to "true" and "false" for the traits. The latter is probably preferable, since it would be nice to see which properties do not depend on the underlying set.

Below are the spaces that appear in Counterexamples along with their number and some note about some variables that affect the traits.

5.) Partition Topology: Depends heavily on partition
7. Deleted Integer Topology: A specific version is in Pi-Base (perhaps this definition is okay, but then where did the data come from?)
12. Closed Extension Topology: Depends at least on cardinality.
16. Open Extension Topology: Depends at least on cardinality. Appears in Pi-Base, but with no traits.
32. Special Subsets of the Real Line: This is actually 10 different subsets with the subspace topology, so it should be broken into separate entries
33. Special Subsets of the Plane: This is actually 2 different subsets with the subspace topology, so it should be broken into separate entries
34. One Point Compactification Topology: Depends heavily on the set
39. Order Topology: Depends at least on order type
44. Uncountable Discrete Ordinal Space: Depends on limit ordinal
49. Right Order Topology: Depends at least on order type
110. Stone-Cech Compactification: This space is traditionally difficult for me to understand, but it must certainly depend on the starting topology
124. Bernstein's Connected Sets: I don't fully understand this example. It appears to be the subspace topology defined on two complicated subsets of the plane. Even if the topology here is definite, there is no information about it in the general reference chart.
134. Bounded Metrics: Depends at least on the starting metric space
137. Cauchy Completion: Depends at least on the starting metric space
138. Hausdorff's Metric Topology: Depends at least on the starting metric space

[jamesdabbs]

Adding other value semantics will take some work. For now, it'd be best to create specific examples where we can (so "uncountable" => "of cardinality \omega_1").