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
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“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)