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AdrianZ
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Is there a method or an algorithm or a theorem or whatever that tells us a topological space is not a Hausdorff space?
AdrianZ said:for example Zariski topology, How do we show that it is non-Hausdorff? I'm just interested to know how we could see if a space is Hausdorff or not.
A topological space is a mathematical concept that describes the properties of a set and its subsets. It is important because it allows us to understand the structure of a space and how its elements are related to each other.
The Hausdorff property is defined as follows: for any two distinct points in a topological space, there exist disjoint open sets containing each point.
To prove that a topological space is non-hausdorff, you must find two distinct points in the space that do not have disjoint open sets containing them. This can be done by constructing a counterexample or by showing that the space violates one of the properties of the Hausdorff property.
Some common examples of non-hausdorff spaces include the line with two origins, the cofinite topology, and the Zariski topology.
No, a topological space cannot be both hausdorff and non-hausdorff. The Hausdorff property is a necessary condition for a topological space to be considered hausdorff, so if a space fails this property, it cannot be hausdorff.