Editing Subspace topology (section)

== Preservation of topological properties ==

If a topological space having some [[topological property]] implies its subspaces have that property, then we say the property is '''hereditary'''. If only closed subspaces must share the property we call it '''weakly hereditary'''.

* Every open and every closed subspace of a [[completely metrizable]] space is completely metrizable.
* 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.