What is the meaning of Hausdorff space?

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What is the meaning of Hausdorff space?​

Hausdorff space. In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where for any two distinct points there exists a neighbourhood of each which is disjoint from the neighbourhood of the other.

Are compact sets Hausdorff spaces?​

For non-Hausdorff spaces, it can be that all compact sets are closed sets (for example, the cocountable topology on an uncountable set) or not (for example, the cofinite topology on an infinite set and the Sierpiński space ). The definition of a Hausdorff space says that points can be separated by neighborhoods.
A topological space is Hausdorff if and only if it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.
What is the difference between Tychonoff and Hausdorff space?
The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff. The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces.

Who is Ben Hausdorff?
Kirstin Maldonado’s to-be husband, Ben Hausdorff, is mostly known as Hasudo. What are they up to? Stay tuned to find out. Ben Hausdorff is a professional photo/videographer.

What is the Hausdorff condition for topology?​

In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions. A simple example of a topology that is T 1 but is not Hausdorff is the cofinite topology defined on an infinite set .

Is Y Hausdorff if ker(f) is closed?​

If f is continuous and Y is Hausdorff then ker ( f) is closed. If f is an open surjection and ker ( f) is closed then Y is Hausdorff. If f is a continuous, open surjection (i.e. an open quotient map) then Y is Hausdorff if and only if ker (f) is closed.

Main proof: Let T be a compact Hausdorff topological space and let X and Y be disjoint closed sets in T. By the lemma, for any y ∈ Y there exists a neighborhood U y ∋ y and an open set O y containing X such that U y ∩ O y = ∅.