classification of topological properties according to behaviour under mapping

Topological properties may be classified by their behaviour with respect to mappings. The basis of such a classification is the following question: Given two topological spaces $X$ and $Y$ and a continuous map $f:X\to Y$, can one infer that one of the spaces has a certain topological property from the fact that the other space has this property?

A trivial case of this question may be disposed of. If $f$ is a homeomorphism, then the spaces $X$ and $Y$ cannot be distinguished using only the techniques of topology, and hence both spaces will have exactly the same topological properties.

To obtain a non-trivial classification, we must consider more general maps. Since every map may be expressed as the composition of an inclusion and a surjection, it is natural to consider the cases where $f$ is an inclusion and where it is a surjection.

In the case of an inclusion, we can define the following classifications:

A property of a topological space is called hereditary if it is the case that whenever a space has that property, every subspace of that space also has the same property.

A property of a topological space is called weakly hereditary if it is the case that whenever a space has that property, every closed subspace of that space also has the same property.

In the case of a surjection, we can define the following classifications:

A property of a topological space is called continuous if it is the case that, whenever a space has this property, the images of this space under all continuous mapping also have the same property.

A property of a topological space is called open if it is the case that, whenever a space has this property, the images of this space under all open continuous mappings also have the same property.

A property of a topological space is called closed invariant if it is the case that, whenever a space has this property, the images of this space under all closed continuous mapping also have the same property.

Title	classification of topological properties according to behaviour under mapping
Canonical name	ClassificationOfTopologicalPropertiesAccordingToBehaviourUnderMapping
Date of creation	2013-03-22 14:38:04
Last modified on	2013-03-22 14:38:04
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	15
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 54C05
Defines	hereditary
Defines	hereditarily
Defines	weakly hereditary
Defines	continuous
Defines	open
Defines	closed invariant