Properties
The subspace topology has the following characteristic property. Let be a subspace of and let be the inclusion map. Then for any topological space a map is continuous if and only if the composite map is continuous.
This property is characteristic in the sense that it can be used to define the subspace topology on .
We list some further properties of the subspace topology. In the following let be a subspace of .
- If is continuous the restriction to is continuous.
- If is continuous then is continuous.
- The closed sets in are precisely the intersections of with closed sets in .
- If is a subspace of then is also a subspace of with the same topology. In other words the subspace topology that inherits from is the same as the one it inherits from .
- Suppose is an open subspace of . Then a subspace of is open in if and only if it is open in .
- Suppose is a closed subspace of . Then a subspace of is closed in if and only if it is closed in .
- If is a base for then is a basis for .
- The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.
Read more about this topic: Subspace Topology
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—Ralph Waldo Emerson (1803–1882)
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