Properties
An important example of these spaces would be, assuming Axiom of Choice, the linearly ordered spaces; however, it really needs axiom of choice for an arbitrary linear order to be normal (see van Douwen's paper). Any generalised metric is monotonically normal even without choice. An important property of monotonically normal spaces is that any two separated subsets are strongly separated there. Monotone normality is hereditary property and a monotonically normal space is always normal by the first condition of the second equivalent definition.
We list up some of the properties :
- A closed map preserves monotone normality.
- A monotonically normal space is hereditarily collectionwise normal.
- Elastic spaces are monotonically normal.
Read more about this topic: Monotonically Normal Space
Famous quotes containing the word properties:
“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)
“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)