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.

[prabau]

The references quoted in Encyclopedia of Gen Top are also listed in the dantopology post. So adding Encycl of GT does not seem necessary.

[Moniker1998]

@prabau it's not about what's necessary but about what's more official

[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.

[prabau]

I think either of these formulations is pretty confusing. Continuing to discuss this in the main thread.

[prabau]

General discussion about meta-properties related to "union".

In general, given a property P, several of the meta-props are phrased as "This property is preserved by such and such a construction". Typically we are given a topological space (or a collection of them) which has the property P, and out of that we construct another space, which then also has the property.

So for "union", what do you want to mean here by "preserved by union"? At first glance, it does not quite fit the above paragraph? Maybe it would be better to rephrase things in a different way.
Can you elaborate the intended concept?

[Moniker1998]

@prabau that if $X = \bigcup_i X_i$ and $X_i$ have P (as subspaces), then $X$ has P

I think this is clear, and we don't need to always talk about constructions. Just because something has seemed like a theme so far, doesn't mean it is.

[prabau]

Exactly! So let's phrase it that way, and not in terms of "constructions that preserve a property".

Something like this for example:
"If $X$ is covered by countably many subspaces, each having the property, then so does $X$."

[Moniker1998]

I am not confused by this, but if you think it's confusing, feel free to standarize your own version, here and in the wiki.

[prabau]

Talking about "preserved by union" is very confusing. One reason is that, apart from the topological disjoint union (= categorical coproduct) of spaces, there is also the notion of (non-disjoint) union, where one starts with a collection of spaces, not necessarily disjoint, and one constructs a new space that is the union of all the spaces as a set, with a certain topology that satisfies some universal property. That construction is not what is meant here. So "preserved by union" is a misleading choice.

[Moniker1998]

Talking about "preserved by union" is very confusing. One reason is that, apart from the topological disjoint union (= categorical coproduct) of spaces, there is also the notion of (non-disjoint) union, where one starts with a collection of spaces, not necessarily disjoint, and one constructs a new space that is the union of all the spaces as a set, with a certain topology that satisfies some universal property. That construction is not what is meant here. So "preserved by union" is a misleading choice.

No because that construction has a name. I don't see a reason to continue this discussion though, we've already agreed that we can do what you prefer.

[felixpernegger]

I am not confused by this, but if you think it's confusing, feel free to standarize your own version, here and in the wiki.

I also think it is unclear what is meant by This property is preserved by countable unions.

"If X is covered by countably many subspaces, each having the property, then so does X ."

this sounds good to me