Semi-regular space

A semi-regular space is a mathematical object from the set-theoretical topology . It is a generalization of regular space , whose regular open subsets form a basis .

definition

A topological space is called semi-regular if the regular open subsets form a basis of the space . A subset of a topological space is called regularly open if and only if the interior of its closure is. That is, it is regularly open if and only if applies. Regularly open sets are also called canonically open sets. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G=\operatorname {int} (\operatorname {cl} (G))}$

properties

• All regular open subsets of a topological space together with the partial order and the regular set operations , , a complete Boolean algebra . ${\displaystyle \subseteq }$ ${\displaystyle \cap ^{\ast }}$ ${\displaystyle \cup ^{\ast }}$ ${\displaystyle \mathrm {C} ^{\ast }}$
• Every regular room is also semi- regular . In particular, the regularly open subsets form a basis of , but not all topological spaces whose regularly open subsets form a basis are regular. ${\displaystyle X}$ ${\displaystyle X}$
• Any topological space can be embedded in a semi-regular space . To do this, consider the set , which is a closed unit interval , and explain a topology on it. The open sets of this topology are given for with for small positive through . And for they are given through , whereby an open environment of for all and is small and positive. This space is itself semi-regular and is embedded as a closed, nowhere dense subspace . ${\displaystyle X}$ ${\displaystyle X\times I}$ ${\displaystyle I}$ ${\displaystyle [0,1]}$ ${\displaystyle (x,y)\in X\times I}$ ${\displaystyle y\neq 0}$ ${\displaystyle \epsilon }$ ${\displaystyle \{(x,z):y-\epsilon <z<y+\epsilon \}}$ ${\displaystyle (x,0)\in X\times I}$ ${\displaystyle \{(x',z):x'\in U,0\leq z<\epsilon _{U}\}}$ ${\displaystyle U}$ ${\displaystyle x\in X}$ ${\displaystyle x'\in U}$ ${\displaystyle \epsilon _{U}}$ ${\displaystyle X}$
• From the third property it can be seen that subspaces of semi-regular spaces are generally not semi-regular.

literature

• Stephen Willard: General Topology. Dover Publications, Mineola NY et al. 2004, ISBN 0-486-43479-6 , chap. 3D & 14E.

Individual evidence

1. ↑ ^{a b} Pavel S. Aleksandrov : Textbook of set theory. 7th edition. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1657-8 , p. 122.
2. ↑ ^{a b} Lothar Ridder: Mereology. A contribution to ontology and epistemology (= Philosophical Treatises. Vol. 83). Klostermann, Frankfurt am Main 2002, ISBN 3-465-03168-7 , p. 170.