Properties
Every first countable hemicompact space is locally compact.
If is a hemicompact space, then the space of all continuous functions to a metric space with the compact-open topology is metrizable. To see this, take a sequence of compact subsets of such that every compact subset of lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of ). Denote
for and . Then
defines a metric on which induces the compact-open topology.
Read more about this topic: Hemicompact 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)