Has closed discrete subset of size 𝔠 (part 4)

[yhx-12243]

Continuation of #1584, #1592 and #1597.

[Moniker1998]

Let's add a meta-property for P198 about unions.

[Moniker1998]

Could cite Encyclopedia of General Topology instead since that's what the link uses.

[Moniker1998]

We could use the meta-property instead

[prabau]

P227: "If a closed subspace of $X$ satisfies this property, so does $X$."
That's a true fact. But I am not sure we need to add this as a meta-property. It does not quite fit into the mold of other meta-properties, and it's also kind of immediate. If we need that to justify some trait, we can just do it directly without referring to this as a meta-property.

(For comparison, consider the property "cardinality at least 4". It would be true to say: "If a subspace of $X$ has the property, so does $X$." But it's not very helpful, it's just too trivial.)

At least, let me review the rest to see how it is used.

[Moniker1998]

P227: "If a closed subspace of X satisfies this property, so does X ."
That's a true fact. But I am not sure we need to add this as a meta-property. It does not quite fit into the mold of other meta-properties, and it's also kind of immediate. If we need that to justify some trait, we can just do it directly without referring to this as a meta-property.

I disagree. We should collect meta-properties when we can.

(For comparison, consider the property "cardinality at least 4". It would be true to say: "If a subspace of X has the property, so does X ." But it's not very helpful, it's just too trivial.)

We can add it once we're using it in an argument.

[prabau]

Let's see what other people have to say.

[prabau]

Q: What that meant to be the topological disjoint union or just any union of subspaces?

[yhx-12243]

Any union. P198 should be preserved by any countable unions. I'll add to meta-properties.

[yhx-12243]

P227: "If a closed subspace of X satisfies this property, so does X ." That's a true fact. But I am not sure we need to add this as a meta-property. It does not quite fit into the mold of other meta-properties, and it's also kind of immediate. If we need that to justify some trait, we can just do it directly without referring to this as a meta-property.

(For comparison, consider the property "cardinality at least 4". It would be true to say: "If a subspace of X has the property, so does X ." But it's not very helpful, it's just too trivial.)

Basically, this is what I called “reverse hereditary” before.

And the former discussion about how to say “reverse hereditary” is exactly “If a closed subspace of X satisfies this property, so does X.”

[prabau]

Ok. If both of you think it's valuable, I won't object.

[prabau]

see #1242

[Moniker1998]

Suggested change

	- This property is preserved by countable (not necessarily disjoint) unions.
	- This property is preserved by countable unions.