Meta-property: reverse heredity

[yhx-12243]

Many property has a meta-property that like a reverse heredity:

$X$ has $P$, if $X$ contains a dense/open/closed subspace that is $P$.

Equivalently,

If, $X$ has $\neg P$, then each dense/open/closed subspace of $X$ is $\neg P$.

For example, P36 (Connectedness), P189 ($\sigma$-connectedness), P64 (Baire) are all example that is reversely hereditary to dense sets.

How to describe these? @prabau

[prabau]

Good question. I agree that it would be valuable to add this in suitable cases. And "reversely hereditary" does not seem a good name.

Here is one way to express this in a generic way without having to explicitly name/link the property (since the property is the same for the subspace and the full space):

"If a dense subspace of $X$ satisfies this property, so does $X$."

Compare for example with https://topology.pi-base.org/properties/P000013:
"$X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does."

[pzjp]

Possibly the statement the property extends onto the closure of the set is worth consideration?

[prabau]

Possibly the statement the property extends onto the closure of the set is worth consideration?

If "extends onto the closure of the set" is supposed to mean the same thing, it seems a little unusual way to say it. But we could use a more direct formulation, for example:

"The closure of a $\sigma$-connected set is $\sigma$-connected."

(with the implied understanding that we are talking about subsets of a given topological space with the subspace topology; so that is equivalent to the other formulation in terms of dense subspaces)