If Y is a subset of X, then Y inherits a total order from X. Y therefore has an order topology, the induced order topology. As a subset of X, Y also has a subspace topology. The subspace topology is always at least as fine as the induced order topology, but they are not in general the same.
For example, consider the subset Y = {–1} ∪ {1/n}n∈N in the rationals. Under the subspace topology, the singleton set {–1} is open in Y, but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space.
Read more about this topic: Order Topology
Famous quotes containing the words induced and/or order:
“The classicist, and the naturalist who has much in common with him, refuse to see in the highest works of art anything but the exercise of judgement, sensibility, and skill. The romanticist cannot be satisfied with such a normal standard; for him art is essentially irrational—an experience beyond normality, sometimes destructive of normality, and at the very least evocative of that state of wonder which is the state of mind induced by the immediately inexplicable.”
—Sir Herbert Read (1893–1968)
“A wise man should order his interests, and set them all in their proper places. This order is often troubled by greed, which puts us upon pursuing so many things at once that, in eagerness for matters of less consideration, we grasp at trifles, and let go things of greater value.”
—François, Duc De La Rochefoucauld (1613–1680)