Size Function - Main Properties

Main Properties

Assume that is a compact locally connected Hausdorff space. The following statements hold:

¤ every size function is a non-decreasing function in the variable and a non-increasing function in the variable .

¤ every size function is locally right-constant in both its variables.

¤ for every, is finite.

¤ for every and every, .

¤ for every and every, equals the number of connected components of on which the minimum value of is smaller than or equal to .

If we also assume that is a smooth closed manifold and is a -function, the following useful property holds:

¤ in order that is a discontinuity point for it is necessary that either or or both are critical values for .

A strong link between the concept of size function and the concept of natural pseudodistance between the size pairs exists ,

¤ if then .

The previous result gives an easy way to get lower bounds for the natural pseudodistance and is one of the main motivation to introduce the concept of size function.

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