Hausdorff space

Hausdorff space is a topological space satisfying the strong separability axiom T ₂ . Named after Felix Hausdorff - one of the founders of the general topology . His initial definition of a topological space included a requirement, which is now called Hausdorff. Sometimes the term Hausdorff topology is used to denote the structure of a Hausdorff topological space on a set.

Content

Definition

Topological space $X$ ${\ displaystyle X}$ called Hausdorff if any two different points $x$ ${\ displaystyle x}$ , $y$ ${\ displaystyle y}$ of $X$ ${\ displaystyle X}$ disjoint surroundings $U(x)$ ${\ displaystyle U (x)}$ , $V(y)$ ${\ displaystyle V (y)}$ .

Examples and counterexamples

Hausdorff spaces are all metric spaces and metrizable spaces , in particular: Euclidean spaces $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ , manifolds , most used in the analysis of infinite-dimensional functional spaces, such as $L^{p}\$ ${\ displaystyle L ^ {p} \}$ or $W^{1,\;p}$ ${\ displaystyle W ^ {1, \; p}}$ , $p\geqslant 1\$ ${\ displaystyle p \ geqslant 1 \}$ .

If a topological group is a T ₀ -space , then it is Hausdorff. If T _{0 is} not satisfied, then factorization by the closure of the neutral element of the group will give a Hausdorff space ^[1] . For this reason, some sources include Hausdorff in the definition of a topological group.

The simplest (and important) example of a non-Hausdorff space is a connected colon , and in the more general case, Heyting algebras . It is not Hausdorff, for example, the Zariski topology on an algebraic variety. Nekhausdorf, generally speaking, the spectrum of a ring .

Properties

The uniqueness of the sequence limit (in the more general case, the filter ), if such a limit exists.
A property equivalent to the definition of the Hausdorff topology is the closedness of the diagonal $\Delta =\{(x,\;x)\;|\;x\in X\}$ ${\ displaystyle \ Delta = \ {(x, \; x) \; | \; x \ in X \}}$ in a Cartesian square $X\times X$ ${\ displaystyle X \ times X}$ of space $X$ ${\ displaystyle X}$ .
In a Hausdorff space, all its points are closed (i.e., single-point sets).
The subspace and Cartesian product of Hausdorff spaces are also Hausdorff.
Generally speaking, Hausdorff cannot be transmitted to quotient spaces .
A compact Hausdorff space is normal and it is metrizable if and only if it has a countable base of topology.
Any continuous one-to-one mapping of a compact space into a Hausdorff space is a homeomorphism .
Any finite Hausdorff space is discrete.

Notes

↑ D. Ramakrishnan and R. Valenza. Fourier Analysis on Number Fields. - Springer-Verlag, 1999. - (Graduate Texts in Mathematics).

Literature

Aleksandrov P. S. Introduction to set theory and general topology. - 2nd, stereotyped. - M .: Doe, 2010 .-- 368 p. - ISBN 978-5-8114-0981-5 .

[1] D. Ramakrishnan and R. Valenza. Fourier Analysis on Number Fields. - Springer-Verlag, 1999. - (Graduate Texts in Mathematics).