Properties
- A topological space is compact if and only if the map from that space to a single point is proper.
- Every continuous map from a compact space to a Hausdorff space is both proper and closed.
- If f : X → Y is a proper continuous map and Y is a compactly generated Hausdorff space (this includes Hausdorff spaces which are either first-countable or locally compact), then f is closed.
Read more about this topic: Proper Map
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)