Meta-properties: preservation under certain maps

[prabau]

I think it would be worthwhile to add some meta-properties having to do with preservation under certain types of maps. In particular:

• compact and connected are preserved by continuous maps
• T4 and T6 are preserved by closed maps (Engelking 1.5.20).

So what would be the best way to express this?

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For comparison, Engelking (p. 34) uses the terminology "property P is invariant of (some class of mappings)" to mean: if $f:X\to Y$ is a surjective map of that class and X has the property, then so does Y.
(surjective is important here, and is implicitly assumed; properties would not be preserved in general if the map is not onto)
So Engelking uses things like "invariant of continuous mappings", "invariant of open mappings", etc.

Another thing to specify is the meaning of open/closed map. Engelking (and Encycl. or Gen Top ?) assumes open/closed maps are continuous (def. p. 31). Other sources don't (wikipedia, Willard, Lee, etc). To be unambiguous, we probably should say "continuous open/closed map" if continuity is assumed.

Now some possible pi-base suggestions for various meta-properties:

• preserved by quotient maps (already in our list, used for P79)
• preserved by continuous maps
• preserved by continuous closed maps
• preserved by continuous open maps

Thoughts?

[felixpernegger]

why not