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Pippi
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I am trying to understand the difference between ordered topology and subspace topology. For one, how do I write down ordered topology of the form {1} x (1, 2] ? How do I write down a basis for {1,2} x Z_+ ?
An ordered topology is a mathematical concept that describes a particular way of ordering elements in a set. In this context, it refers to the way that the real numbers on the x-axis and y-axis of a Cartesian plane are ordered and how this affects the topology of the space.
The topology of R x R, also known as the product topology, is a topology on the Cartesian product of two topological spaces. In this case, R x R refers to the Cartesian plane, and the topology is defined by open sets that are the unions of intervals along the x-axis and y-axis.
The usual topology on R is defined by open intervals, while the topology of R x R is defined by open sets that are unions of intervals along both the x-axis and y-axis. This means that in the product topology, open sets can be more complex and contain sets that are not intervals, such as the union of two intersecting intervals.
Some examples of open sets in the ordered topology on R x R include:
The ordered topology on R x R is closely related to the concept of ordered pairs, as it is defined by the ordering of elements on the x-axis and y-axis in a Cartesian plane. In this topology, the order of the elements in a pair matters, as it will determine the position of the point on the Cartesian plane and which open sets it belongs to. This is different from other topologies where the order of elements in a pair may not matter, such as in the discrete topology.