Mysior level

The Mysior plane is an example of a topological space from the year 1981 going back to the Polish mathematician Adam Mysior . It is a regular Hausdorff space that is not completely regular , or, expressed in separation axioms , a T ₃ space that is not T _3a space is. The construction is much simpler than older examples of this type.

definition

The zero environments in the Mysior plane

The basic amount of the space presented here is the upper half level together with another point that can be chosen as . ${\displaystyle P}$ ${\displaystyle (0,-1)}$

${\displaystyle X=\mathbb {R} \times \mathbb {R} _{0}^{+}\cup \{(0,-1)\}}$.

The topology is defined by specifying surrounding bases. We consider the base of a point to be: ${\displaystyle (x,y)\in X}$

• in the case of the set , that is, these points should all be isolated . ${\displaystyle y>0}$ ${\displaystyle \{(x,y)\}}$
• in the case of the set of sets of the form , where lies in the union of the lines and and, with a maximum of finitely many exceptions, also contains all points from . ${\displaystyle y=0}$ ${\displaystyle \{(x,0)\}\cup S}$ ${\displaystyle S}$ ${\displaystyle I_{x}:=\{x\}\times [0,2)}$ ${\displaystyle J_{x}:=\{(x+\xi ,\xi );0\leq \xi <2\}}$ ${\displaystyle I_{x}\cup J_{x}}$
• in the case of the quantity of quantities , this case only concerns the point . ${\displaystyle y=-1}$ ${\displaystyle U_{n}:=\{(\xi ,\eta );\xi >n,\eta \geq 0\}\cup \{(0,-1)\}}$ ${\displaystyle P=(0,-1)}$

A topology is defined by these environment bases. The topological space is called the Mysior level. ${\displaystyle \tau }$ ${\displaystyle X}$ ${\displaystyle (X,\tau )}$

properties

The point is not isolated, even if the above sketch gives this impression, because it obviously converges for . ${\displaystyle P=(0,-1)}$ ${\displaystyle (n,0)\rightarrow P}$ ${\displaystyle n\to \infty }$

The Mysior level is a T ₃ room . The Hausdorff property , according to which two points have disjoint neighborhoods, can easily be read from the specified neighborhood bases. The space is also regular , which means that every point has an environment base of closed sets. In the first two cases of the above definition, the specified quantities have already been completed. For the neighborhood basis amounts of changed her one , so there is also a neighborhood basis of closed sets. ${\displaystyle (X,\tau )}$ ${\displaystyle \tau }$ ${\displaystyle U_{n}}$ ${\displaystyle P}$ ${\displaystyle {\overline {U_{n+2}}}\subset U_{n}}$

The Mysior level is not a T _3a room . The set is closed , is a point outside of this set, but one can show that every continuous function that satisfies for all is also 0 at the point . In this technical part, use is made of the structure of the surrounding bases of the points , with which zeros can be "transported to the right" in such a way that there are infinitely many zeros in each interval . Thus every neighborhood of zeros of and from contains the continuity of follows . Therefore there can not be a T _3a space. ${\displaystyle A:=(-\infty ,1]\times \{0\}\subset X}$ ${\displaystyle \tau }$ ${\displaystyle P\in X}$ ${\displaystyle f(a)=0}$ ${\displaystyle a\in A}$ ${\displaystyle P}$ ${\displaystyle (x,0)}$ ${\displaystyle f}$ ${\displaystyle [n,n+1]\times \{0\}}$ ${\displaystyle U_{n}}$ ${\displaystyle P}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f(P)=0}$ ${\displaystyle (X,\tau )}$

Individual evidence

1. ^ Adam Mysior: A Regular Space which is not Completely Regular. In: Proceedings of the American Mathematical Society. Vol. 81, No. 4, 1981, pp. 652-653, doi : 10.1090 / S0002-9939-1981-0601748-4 .
2. ↑ Jürgen Heine: Topology and Functional Analysis. Basics of abstract analysis with applications. Oldenbourg, Munich et al. 2002, ISBN 3-486-24914-2 , example (2.5,4).
3. ↑ Jun-iti Nagata : Modern General Topology (= North Holland Mathematical Library. Vol. 33). 2., revised edition. North-Holland, Amsterdam et al. 1985, ISBN 0-444-87655-3 , Example III.2.