When is the Order Topology the Discrete Topology? Conditions
The order topology on a partially ordered set is the topology generated by the open intervals of the set. The discrete topology on a set is the topology where every subset of the set is open.
The order topology is the discrete topology if and only if the partially ordered set is totally ordered.
Proof:
Suppose the partially ordered set is totally ordered. Then every open interval in the order topology is of the form $(a, b)$ or $[a, b)$, where $a$ and $b$ are elements of the set. Since every subset of a totally ordered set is either an open interval or a union of open intervals, the order topology is the discrete topology.
Conversely, suppose the order topology is the discrete topology. Then every subset of the set is open. In particular, every open interval is open. This implies that the set is totally ordered.
Related Questions:
- What is the order topology?
- The order topology is the topology generated by the open intervals of a partially ordered set.
- What is the discrete topology?
- The discrete topology is the topology where every subset of the set is open.
- When is the order topology the discrete topology?
- The order topology is the discrete topology if and only if the partially ordered set is totally ordered.
- Is the order topology on the set of real numbers the discrete topology?
- No, the set of real numbers is not totally ordered, so the order topology is not the discrete topology.
- Is the order topology on the set of rational numbers the discrete topology?
- Yes, the set of rational numbers is totally ordered, so the order topology is the discrete topology.
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