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
Famous quotes containing the word properties:
“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)
“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)