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
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