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.
Read more about this topic: Size Function
Famous quotes containing the words main and/or properties:
“Parents need to recognize that the negative behavior accompanying certain stages is just a small part of the total child. It should not become the main focus or be pushed into the limelight.”
—Saf Lerman (20th century)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)