Trivial topology
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Definition
The trivial topology (sometimes known as the indiscrete topology[1])is an example of a topological space that exists for any set [ilmath]X[/ilmath], it is defined as follows[1]:
- Given a set [ilmath]X[/ilmath] we define the open sets as [ilmath]\mathcal{J}:=\{\emptyset,X\}[/ilmath]
Then [ilmath](X,\mathcal{J})[/ilmath] is a topology.
Contrast to the Discrete topology
There is at least 1 other topology that can be defined on an arbitrary set, the Discrete topology, which is a topology induced by a metric, the Discrete metric specifically.
Warning, the following is Alec's speculation
- Unlike the discrete topology the indiscrete, or trivial topology is not induced by a metric. For if such a metric existed it would have to have the open ball of radius [ilmath]0[/ilmath] as the entire of [ilmath]X[/ilmath], then no strict :subset of [ilmath]X[/ilmath] (except the emptyset) is a neighborhood to all of its points, thus not open.
- However this is not a metric as the metric must assign two points a distance of [ilmath]0[/ilmath] if and only if they are the same point.
- I'm not entirely happy about this proof, however there is logic here! I will do this formally later.
TODO: Formally prove this
End of warning
