Preservation of Topological Properties
If whenever a topological space has a certain topological property we have that all of its subspaces share the same property, then we say the topological property is hereditary. If only closed subspaces must share the property we call it weakly hereditary.
- Every open and every closed subspace of a topologically complete space is topologically complete.
- Every open subspace of a Baire space is a Baire space.
- Every closed subspace of a compact space is compact.
- Being a Hausdorff space is hereditary.
- Being a normal space is weakly hereditary.
- Total boundedness is hereditary.
- Being totally disconnected is hereditary.
- First countability and second countability are hereditary.
Read more about this topic: Subspace Topology
Famous quotes containing the words preservation of, preservation and/or properties:
“There is something to be said for jealousy, because it only designs the preservation of some good which we either have or think we have a right to. But envy is a raging madness that cannot bear the wealth or fortune of others.”
—François, Duc De La Rochefoucauld (1613–1680)
“The bourgeois treasures nothing more highly than the self.... And so at the cost of intensity he achieves his own preservation and security. His harvest is a quiet mind which he prefers to being possessed by God, as he prefers comfort to pleasure, convenience to liberty, and a pleasant temperature to that deathly inner consuming fire.”
—Hermann Hesse (1877–1962)
“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)