Properties
- (Alexandrov's theorem) If X is Polish then so is any Gδ subset of X.
- (Cantor–Bendixson theorem) If X is Polish then any closed subset of X can be written as the disjoint union of a perfect subset and a countable subset.
The converse of Alexandrov's theorem is true as well: if a subspace S of a Polish space X is Polish, then it is a Gδ subset of X.
Read more about this topic: Polish 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)