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:
“Observe this, that tho’ a woman swear, forswear, lie, dissemble, back-bite, be proud, vain, malicious, anything, if she secures the main chance, she’s still virtuous; that’s a maxim.”
—George Farquhar (1678–1707)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)