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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