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Oxymoron
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Can someone provide me with an example of a non-Hausdorff space. I can't seem to conjure one up
Thus in the indsicrete topology every sequence tends to every point since there is only one open set (apart from the empty set).
Oxymoron said:So open sets in X are determined by the topology on X?
And a sequence [itex]x_n[/itex] converges to a point [itex]x[/itex] if each open neighbourhood of [itex]x[/itex] contains [itex]x_n [/itex] for [itex]n[/itex] sufficiently large.
So does that mean if a sequence exists in the indiscrete topological space, then the elements of the sequence must reside in an open set containing [itex]x_n[/itex].
And the only open set containing any points is [itex]X[/itex] because, from the topology, every other set is not open, or empty.
Fortunately, I'm already sitting down! If X is any set, the definition of "topology on X" is a collection of subsets of X such that:Oxymoron said:I just want to ask. Because we have defined the topology to be indiscrete, that means that the only open sets are the empty set and itself right? So open sets in X are determined by the topology on X?
? In any topology a point of a sequence is in some open set! It does happen that, in the discrete topology, the only open set is X itself so the entire sequence is in that set!And a sequence [itex]x_n[/itex] converges to a point [itex]x[/itex] if each open neighbourhood of [itex]x[/itex] contains [itex]x_n [/itex] for [itex]n[/itex] sufficiently large. So does that mean if a sequence exists in the indiscrete topological space, then the elements of the sequence must reside in an open set containing [itex]x_n[/itex].
And the only open set containing any points is [itex]X[/itex] because, from the topology, every other set is not open, or empty.
A non-Hausdorff space is a type of topological space in which there exist points that cannot be separated by disjoint open sets. This means that there are points in the space that cannot be isolated from each other by open sets, unlike in a Hausdorff space where all points can be separated by open sets.
Non-Hausdorff spaces have many applications in mathematics and other fields, such as physics and computer science. They can be used to model more complex and non-intuitive spaces, and can also be used to study and understand the behavior of systems that exhibit non-Hausdorff properties.
One example of a non-Hausdorff space is the Zariski topology on an algebraic variety. In this topology, the closed sets are defined as the zero sets of collections of polynomials. This space is non-Hausdorff because there exist points that cannot be separated by disjoint open sets.
In a non-Hausdorff space, there exist points that cannot be separated by disjoint open sets, while in a Hausdorff space, all points can be separated in this way. Additionally, non-Hausdorff spaces do not satisfy the T2 separation axiom, which states that any two distinct points in a space can be separated by open sets. Hausdorff spaces, on the other hand, do satisfy this axiom.
Yes, there are many real-world examples of non-Hausdorff spaces. Some examples include the spaces of continuous functions with the compact-open topology, the space of all measurable functions with the L1 topology, and the space of all compact subsets of a topological space with the Hausdorff metric. These spaces have applications in fields such as functional analysis and topology.