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# 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 $f\colon X\to Y$, 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.

## Mathematics Subject Classification

54C05*no label found*

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