Axiom Systems of General Topology

General topology deals with topology based on a system of axioms in the context of set theory . It is therefore also called set- theoretical topology . As has been shown, there are a number of equivalent possibilities within this framework to axiomatically define the structure of the topological spaces . A basic set is always assumed, the elements of which are often called points . The set is then also referred to as the point set . The axiomatic defining the topological structure takes place either by the fact that certain subset systems within the associated power set to be excellent, or by way of the setting certain set operators on , in each case the fulfillment of a number of conditions, axioms is required mentioned. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {P}}=\{W\mid W\subseteq X\}}$ ${\displaystyle {\mathcal {P}}}$

Open set, topologies, axioms of open sets

According to today's view, a topological space is understood to be a pair with a set and a subset system of open sets, so that the following axioms apply: ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}\subseteq {\mathcal {P}}}$

(O1)   ${\displaystyle \emptyset \in {\mathcal {O}}}$

(O2)   ${\displaystyle X\in {\mathcal {O}}}$

(O3)   ${\displaystyle \forall {\mathcal {W}}\subseteq {\mathcal {O}}:{\bigcup {\mathcal {W}}}\in {\mathcal {O}}}$

(O4)   ${\displaystyle \forall n\in \mathbb {N} :\forall {W_{1},W_{2},\dotsc ,W_{n}}\in {\mathcal {O}}:{W_{1}\cap W_{2}\cap \dotsb \cap W_{n}}\in {\mathcal {O}}}$

One also calls the system of - open sets . Instead of an - open set , one only speaks of an open set if it can be assumed that it is clear from the context which topological space is involved. ${\displaystyle {\mathcal {O}}}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle (X,{\mathcal {O}})}$

Under this convention, these axioms can also be stated as follows:

(O1) `The empty set is open.

(O2) `The basic quantity is open. ${\displaystyle X}$

(O3) `Any associations open sets are open.

(O4) `Arbitrary finite averages of open sets are open.

The concept of the open set is now considered the basic concept of the axiomatics of topological spaces . Most modern authors understand (Engl. Under a topology topology ) the system of open sets of a topological space . However, there are exceptions.

Closed set, axioms of closed sets, duality

The closed sets of the topology arise from the open sets through complement formation and vice versa. ${\displaystyle (X,{\mathcal {O}})}$

This means:

(OA)   is a - closed set or - according to the convention (see above) - a closed set if and only if an - open subset or open. ${\displaystyle A\subseteq X}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle (X\setminus A)}$ ${\displaystyle (X,{\mathcal {O}})}$

Since the formation of complement now works involutorially on the power set , the axiom system (O1) - (O4) with regard to the system of open sets can be transferred to an equivalent axiom system with regard to the system of closed sets and vice versa. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {A}}=\{X\setminus W\mid W\in {\mathcal {O}}\}}$

This gives us the following four axioms of closed sets:

(A1)   ${\displaystyle X\in {\mathcal {A}}}$

(A2)   ${\displaystyle \emptyset \in {\mathcal {A}}}$

(A3)   ${\displaystyle \forall {\mathcal {A}}_{0}\subseteq {\mathcal {A}}:{\bigcap {{\mathcal {A}}_{0}}}\in {\mathcal {A}}}$

(A4)   ${\displaystyle \forall n\in \mathbb {N} :\forall {A_{1},A_{2},\dotsc ,A_{n}}\in {\mathcal {A}}:{A_{1}\cup A_{2}\cup \dotsb \cup A_{n}}\in {\mathcal {A}}}$

The axiom system (A1) - (A4) can also be expressed in words like this :

(A1) `The basic set   is complete. ${\displaystyle X}$

(A2) `The empty set is complete.

(A3) `Any averages of closed sets are closed.

(A4) `Arbitrary finite unions of closed sets are closed.

If a system of closed sets is given, which fulfills the axiom system (A1) - (A4) , one obtains a system of open sets , i.e. the associated topology , as complements of the closed sets :

(AO)   ${\displaystyle {\mathcal {O}}=\{X\setminus A\mid A\in {\mathcal {A}}\}}$

The axiom systems (O1) - (O4) and (A1) - (A4) are therefore equivalent in a dual sense . That means: The two axiom systems are reversibly and clearly related to one another and linked to one another via the formation of a complement . In this context, one speaks of the duality between open and closed sets .

Completed hull, Kuratovskian hull operator, Kuratovsky's axioms

The access to general topology by means of shell operators goes back to the Polish mathematician Kazimierz Kuratowski . This axiomatic basis is a set operator on , which is excellent in that it for subsets and meets the following four conditions: ${\displaystyle {\mathcal {P}}}$ ${\displaystyle V\subseteq X}$ ${\displaystyle W\subseteq X}$

(AH1)   ${\displaystyle W\subseteq {\overline {W}}}$

(AH2)   ${\displaystyle {\overline {V\cup W}}={\overline {V}}\cup {\overline {W}}}$

(AH3)   ${\displaystyle {\overline {\overline {W}}}={\overline {W}}}$

(AH4)   ${\displaystyle {\overline {\emptyset }}=\emptyset }$

One calls these four conditions axioms Kuratowski or Kuratowskische shell axioms (Engl. Kuratowski closure axioms ) and these conditions sufficient quantity operator a Kuratowskischen closure operator .

Kuratowski's axioms can be summarized as follows:

(AH) `A Kuratovskian hull operator auf is a hull operator which fulfills the conditions (AH2) and (AH4). ${\displaystyle {\mathcal {P}}}$

If a Kuratovskian hull operator is given, one says:

(AH-A)     is a closed set or closed if and only if is. ${\displaystyle W\subseteq X}$ ${\displaystyle {\overline {W}}=W}$

The subset of the system (in this sense) is the closed sets the closure operator associated sheath system and satisfies the above axiom system (A1) - (A4) , and consequently as described above to a topology leads to . The following applies: ${\displaystyle W\mapsto {\overline {W}}}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle X}$

(AH-O)   ${\displaystyle {\mathcal {O}}:=\{W\subseteq X\mid W\cap {\overline {X\setminus W}}=\emptyset \}}$

(AH-A) '   ${\displaystyle {\mathcal {A}}:=\{{\overline {W}}\mid W\subseteq X\}}$

This view can be reversed:

If a topology is given and the subset system that satisfies the axiom system (A1) - (A4), i.e. the system of closed sets of topological space as described , then there is an envelope system and the associated envelope operator is recovered by: ${\displaystyle {\mathcal {O}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {P}}}$

(A-AH)     (   ) ${\displaystyle {\overline {W}}:=\bigcap \{A\in {\mathcal {A}}\mid A\supseteq W\}}$ ${\displaystyle W\subseteq X}$

This hull operator then fulfills the axioms (AH1) - (AH4) , so it is a Kuratovskian hull operator.

In this way the relationship of the Kuratovskian hull operator to the system of closed sets of topological space , and also to the topology, is reversibly unique . ${\displaystyle W\mapsto {\overline {W}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {O}}}$

For a subset , the closed shell is called , sometimes also the end of . Their elements are called points of contact or points of contact of . According to (A-AH) the closed envelope of is the smallest closed superset of within the topological space with regard to the inclusion relation . ${\displaystyle W\subseteq X}$ ${\displaystyle {\overline {W}}}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle {\overline {W}}}$ ${\displaystyle W\subseteq X}$ ${\displaystyle W}$ ${\displaystyle (X,{\mathcal {O}})}$

Interior, core operator, axioms of the interior

Starting obtained from the duality between open and closed sets in transmission of (A-AH) to the topological space associated core operator on by: ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle W\mapsto {W^{\circ }}}$ ${\displaystyle {\mathcal {P}}}$

(O-OK) ( ) ${\displaystyle {W^{\circ }}=\bigcup \{W_{0}\in {\mathcal {O}}|W_{0}\subseteq W\}}$ ${\displaystyle W\subseteq X}$

back.

The kernel operator satisfies the following four axioms for and : ${\displaystyle V}$ ${\displaystyle W}$

(OK1)   ${\displaystyle {W^{\circ }}\subseteq W}$

(OK2)   ${\displaystyle {(V\cap W)}^{\circ }={V^{\circ }}\cap {W^{\circ }}}$

(OK3)   ${\displaystyle {W^{\circ }}^{\circ }={W^{\circ }}}$

(OK4)   ${\displaystyle {X^{\circ }}=X}$

${\displaystyle {W^{\circ }}}$is because of (O-OK) the largest open subset of within the topological space with regard to the inclusion relation . Its elements are called inner points of . Taken together, the inner points of form the set , which is also called the inner or the open core of . ${\displaystyle W}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle {W^{\circ }}}$ ${\displaystyle W}$

The relationships between the core operator and the topology and , the system of closed sets of and finally the associated Kuratovskian hull operator are pairwise reversible and unambiguous and the following applies: ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (X,{\mathcal {O}})}$

(ECO)   ${\displaystyle {\mathcal {O}}=\{{W^{\circ }}\mid W\subseteq X\}}$

(OK-A)   ${\displaystyle {\mathcal {A}}=\{{X\setminus {W^{\circ }}}\mid W\subseteq X\}}$

(AH-OK)     (   ) ${\displaystyle W^{\circ }=X\setminus {\overline {(X\setminus W)}}}$ ${\displaystyle W\subseteq X}$

(OK-AH)     (   ) ${\displaystyle {\overline {W}}=X\setminus {(X\setminus W)^{\circ }}}$ ${\displaystyle W\subseteq X}$

Edge, edge formation operator, axioms of the edge

For a subset of the topological space that is edge (also known as border or boundary designated; English frontier or boundary ) of     given by: ${\displaystyle W}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle W}$

(AH-R)   ${\displaystyle \partial W={\overline {W}}\cap {\overline {(X\setminus W)}}}$

The elements of are called boundary points of . An edge point of is characterized by the fact that it is both the point of contact of   and the point of contact of   . On the other hand, every point of contact of is either an element of or an edge point of , and thus: ${\displaystyle \partial W}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle X\setminus W}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle W}$

(R-AH)     (    ) ${\displaystyle {\overline {W}}=W\cup \partial W\,}$ ${\displaystyle W\subseteq X}$

For the topological space , the formation of the boundary represents a set operator on . This boundary formation operator , which belongs to this , always satisfies the following four rules for subsets and of : ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle (X,{\mathcal {O}})}$

(R1)   ${\displaystyle V\cap W\cap \partial (V\cap W)=V\cap W\cap (\partial V\cup \partial W)}$

(R2)   ${\displaystyle \partial W=\partial (X\setminus W)}$

(R3)   ${\displaystyle \partial {\partial W}\subseteq \partial W}$

(R4)   ${\displaystyle \partial {\emptyset }=\emptyset }$

Starting from the concept of the edge, the entire axiomatics of general topology can now be built up by understanding the four rules (R1) - (R4) as axioms. The structure of the topological space is thus determined unequivocally. The set operator on defined by means of the equation (R-AH) proves to be a Kuratovskian envelope operator and, in connection with (AH-R), is reversibly and uniquely linked to this and thus also to the associated topological space . ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle (X,{\mathcal {O}})}$

The following equations result: ${\displaystyle \partial }$

(RO)   ${\displaystyle {\mathcal {O}}=\{W\subseteq X\mid {W\cap \partial W}=\emptyset \}}$

(RA)   ${\displaystyle {\mathcal {A}}=\{A\subseteq X\mid {\partial A}\subseteq A\}}$

(OK-R)     (    ) ${\displaystyle \partial W={\overline {W}}\setminus W^{\circ }}$ ${\displaystyle W\subseteq X}$

Derivative, derivative operator, axioms of the derivative

The derivative operator   , which   assigns a subset     of     its derivatives   (English derived set )   , is closely linked to the Kuratovskian envelope operator of a topological space     - similar to the boundary formation operator . Instead of the derivative one also speaks of the derivation of     and writes     or     instead of    . ${\displaystyle (X,{\mathcal {O}})}$   ${\displaystyle d}$ ${\displaystyle W}$ ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle d(W)}$ ${\displaystyle W}$ ${\displaystyle W^{\prime }}$ ${\displaystyle W^{\rm {d}}}$ ${\displaystyle d(W)}$

For a subset of     the Derived   from     equal to the amount of their accumulation points (English accumulation points ), that can be divided into formulas represented as: ${\displaystyle W}$   ${\displaystyle d(W)}$ ${\displaystyle W}$

(AH-D)   ${\displaystyle d(W)=\{x\in X\mid {x\in {\overline {(W\setminus \{x\})}}}\}}$     ${\displaystyle (W\subseteq X)}$

As with the edge of     : ${\displaystyle W}$

(D-AH)   ${\displaystyle {\overline {W}}=W\cup d(W)}$     ${\displaystyle (W\subseteq X)}$

For the topological space     of sufficient quantity operator on     subsets     and     of     always the following four rules: ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle (X,{\mathcal {O}})}$

(D1)   ${\displaystyle d(V\cup W)=d(V)\cup d(W)}$

(D2)   ${\displaystyle d(d(W))\subseteq W\cup d(W)}$

(D3)   ${\displaystyle d(\emptyset )=\emptyset }$

(D4)   ${\displaystyle \forall x\in X:x\notin d(\{x\})}$

Starting from the concept of the derivative and from (D1) - (D4) as a system of axioms , the general topology can be fully developed. Because with this the structure of the topological space is     unequivocally established. The set operator auf   defined by means of the equation (D-AH)   is a Kuratovskian envelope operator and thus, in connection with (AH-D), is reversibly uniquely linked with this and thus also with the associated topological space   . ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {P}}}$   ${\displaystyle (X,{\mathcal {O}})}$

  The following equations result with respect to : ${\displaystyle d}$

(DO)   ${\displaystyle {\mathcal {O}}=\{W\subseteq X\mid {W\cap d(X\setminus W)}=\emptyset \}}$

(THERE)   ${\displaystyle {\mathcal {A}}=\{A\subseteq X\mid d(A)\subseteq A\}}$

(OK-D)   ${\displaystyle d(W)=\{x\in W\mid {x\notin {((X\setminus W)\cup \{x\}})}^{\circ }\}}$     ${\displaystyle (W\subseteq X)}$

Environment, environment filter, environment axioms

The axiomatic structure of the general topology based on the concept of the environment of a point goes back to Felix Hausdorff and his main features of set theory . This classic approach uses environmental systems as the most important structures . Here, each is assigned     a subset   system for which the following rules, called environmental axioms, are assumed to be given: ${\displaystyle x\in X}$   ${\displaystyle {\mathcal {U}}_{x}\subseteq {\mathcal {P}}}$

(U1)   is a filter within     . ${\displaystyle {\mathcal {U}}_{x}}$ ${\displaystyle ({\mathcal {P}},\subseteq )}$

(U2)   ${\displaystyle \forall U\in {\mathcal {U}}_{x}:x\in U}$

(U3)   ${\displaystyle \forall U\in {\mathcal {U}}_{x}:\exists V\in {\mathcal {U}}_{x}:\forall y\in V:U\in {\mathcal {U}}_{y}}$

For   is     also called the environment filter of     and each     an environment of     . It is always     , so     . ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {U}}_{x}}$ ${\displaystyle x}$ ${\displaystyle U\in {\mathcal {U}}_{x}}$ ${\displaystyle x}$ ${\displaystyle X\in {\mathcal {U}}_{x}}$ ${\displaystyle {\mathcal {U}}_{x}\neq \emptyset }$

In a less formalized way, the surrounding axioms can also be expressed in relation to any point   as follows: ${\displaystyle x\in X}$

(U1) `The basic set     is around     . ${\displaystyle X}$ ${\displaystyle x}$

(U2) `     is contained as a point in each of its surroundings. ${\displaystyle x}$

(U3) `Every superset of a neighborhood of     is in turn a neighborhood of    . ${\displaystyle x}$ ${\displaystyle x}$

(U4) `The average of finitely many neighborhoods of     is neighborhood of    . ${\displaystyle x}$ ${\displaystyle x}$

(U5) `Is     neighborhood of     , encompasses     another neighborhood     of     such that     itself belongs to the neighborhoods of every point     . ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle y\in V}$

The structure   described above   is also called the surrounding space. ${\displaystyle (X,({\mathcal {U}}_{x})_{x\in X})}$

Such a surrounding space via     is now clearly reversibly linked to the topological space  , when looking at an in ambient room open set is understood the following: ${\displaystyle X}$   ${\displaystyle (X,{\mathcal {O}})}$

(UO) The subset     is open if and only if it is the neighborhood of each of its points. ${\displaystyle W\subseteq X}$

So:

(UO) '   ${\displaystyle {\mathcal {O}}=\{W\subseteq X\mid {\forall x\in W:W\in {\mathcal {U}}_{x}}\}}$

The environment   filters   belonging to the topological space   can be recovered by: ${\displaystyle (X,{\mathcal {O}})}$   ${\displaystyle {\mathcal {U}}_{x}\subseteq {\mathcal {P}}}$     ${\displaystyle (x\in X)}$

(OU) A subset     is a neighborhood of     then and only if an open subset    , i.e. a    , exists with   . ${\displaystyle U\subseteq X}$ ${\displaystyle x\in X}$ ${\displaystyle W\subseteq X}$ ${\displaystyle W\in {\mathcal {O}}}$ ${\displaystyle x\in W\subseteq U}$

So:

(OU) '   ${\displaystyle {\mathcal {U}}_{x}=\{U\subseteq X\mid {\exists W\in {\mathcal {O}}:x\in W\subseteq U}\}}$

The relationships to the other structural elements are as follows:

- with regard to the closed quantities :

(UA)     is complete if and only if for     the fact that every neighborhood     a   non-empty intersection with     , has already     follows. ${\displaystyle A\subseteq X}$ ${\displaystyle x\in X}$ ${\displaystyle U\in {\mathcal {U}}_{x}}$ ${\displaystyle A}$ ${\displaystyle x\in A}$

So:

(UA) '   ${\displaystyle {\mathcal {A}}=\{A\subseteq X\mid {\forall x\in X:({\forall U\in {\mathcal {U}}_{x}:A\cap U\neq \emptyset }\Rightarrow x\in A})\}}$

- with regard to the Kuratovskian hull operator  :

(U-AH)   ${\displaystyle {\overline {W}}:=\{x\in X\mid \forall U\in {\mathcal {U}}_{x}:{U\cap W\neq \emptyset }\}}$     ${\displaystyle (W\subseteq X)}$  

- with regard to the core operator  :

(U-OK)   ${\displaystyle {W^{\circ }}=\{x\in W\mid \exists U\in {\mathcal {U}}_{x}:U\subseteq W\}}$   ${\displaystyle (W\subseteq X)}$  

- with regard to the edge formation operator  :

(UR)     ${\displaystyle \partial W=\{x\in X\mid \forall U\in {\mathcal {U}}_{x}:{U\cap W\neq \emptyset }\land {U\cap (X\setminus W)\neq \emptyset }\}}$   ${\displaystyle (W\subseteq X)}$  

- with regard to the derivative operator  :

(UD)     ${\displaystyle d(W)=\{x\in X\mid \forall U\in {\mathcal {U}}_{x}:{{(U\setminus \{x\})}\cap W\neq \emptyset }\}}$   ${\displaystyle (W\subseteq X)}$  

literature

• Nicolas Bourbaki : General Topology. Part 1 . Hermann, Paris 1966.
• Thorsten Camps , Stefan Kühling , Gerhard Rosenberger : Introduction to set-theoretical and algebraic topology (=  Berlin study series on mathematics . Volume 15 ). Heldermann Verlag , Lemgo 2006, ISBN 3-88538-115-X .
• James Dugundji : Topology . 8th edition. Allyn and Bacon, Boston 1973.
• Ryszard Engelking: Outline of General Topology . 8th edition. North-Holland Publishing Company, Amsterdam 1968.
• Lutz Führer : General topology with applications . Vieweg Verlag , Braunschweig 1977, ISBN 3-528-03059-3 .
• Egbert Harzheim , Helmut Ratschek: Introduction to General Topology (=  THE MATHEMATICS. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society , Darmstadt 1978, ISBN 3-534-06355-4 . MR0380697
• Egbert Harzheim: Introduction to combinatorial topology (=  THE MATHEMATICS. Introduction to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society , Darmstadt 1978, ISBN 3-534-07016-X . MR0533264
• Felix Hausdorff : Fundamentals of set theory. Reprint. First edition, Berlin, 1914 . Chelsea Publishing Company, Ney York 1965, ISBN 0-8284-0061-X .
• Horst Herrlich: Topology I: Topological Spaces . Heldermann Verlag , Berlin 1986, ISBN 3-88538-102-8 .
• John L. Kelley : General topology. Reprint of the 1955 edition published by Van Nostrand . Springer-Verlag , Berlin / Heidelberg / New York 1975, ISBN 3-540-90125-6 .
• Hans-Joachim Kowalsky : Topological Spaces . Birkhäuser Verlag , Basel et al. 1961.
• Kazimierz Kuratowski : Topology. Volume I . Academic Press, New York et al. 1966.
• Jun-iti Nagata : Modern General Topology (=  North Holland Mathematical Library . Volume 33 ). 2nd, revised edition. North-Holland Publishing Company , Amsterdam, New York, Oxford 1985, ISBN 0-444-87655-3 . MR0831659
• Boto von Querenburg : Set theoretical topology . 3., rework. and exp. Edition. Springer-Verlag , Berlin et al. 2001, ISBN 3-540-67790-9 .
• Gerhard Preuss : General Topology . Springer-Verlag , Berlin et al. 1972, ISBN 3-540-06006-5 .
• Willi Rinow : Textbook of Topology . German Science Publishers, Berlin 1975.
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag , Stuttgart 1975, ISBN 3-519-12200-6 .
• R. Vaidyanathaswamy : Set Topology (Reprint) . 2nd Edition. Chelsea Publishing, Providence, RI 1964.
• Stephen Willard: General Topology . Addison-Wesley , Reading, Massachusetts et al. 1970.

Individual evidence

1. ↑ N. Bourbaki: General Topology . 1966, p. 17 .
2. ↑ T. Camps, S. KühlingG. Rosenberger: Introduction to set theoretical and algebraic topology . 2006, p. 7 .
3. ↑ J. Dugundji: Topology . 1973, p. 62 .
4. ^ R. Engelking: Outline of General Topology . 1968, p. 26 .
5. ↑ L. Guide: General topology with applications . 1977, p. 14 .
6. ↑ H. Herrlich: Topology I: Topological Spaces . 1986, p. 3 .
7. ↑ E. Harzheim, H. Ratschek: Introduction to General Topology . 1978, p. 14 .
8. ↑ J. Nagata: Modern General Topology . 1985, p. 30 .
9. ↑ G. Preuss: General Topology . 1972, p. 21 .
10. ↑ JL Kelley: General topology . 1975, p. 37 .
11. ↑ B. v. Querenburg: Set theoretical topology . 2001, p. 17 .
12. ↑ W. Rinow: Textbook of Topology . 1975, p. 23 .
13. ^ S. Willard: General Topology . 1970, p. 23 .
14. ↑ Kowalsky (p. 41), for example, links the family of environment filters of a topological space with topology .
15. ↑ W. Rinow: Textbook of Topology . 1975, p. 25 .
16. ↑ L. Guide: General topology with applications . 1977, p. 24 .
17. ^ K. Kuratowski: Topology. Volume I . 1966, p. 38 .
18. ↑ W. Rinow: Textbook of Topology . 1975, p. 7 .
19. ^ R. Vaidyanathaswamy: Set Topology . 1964, p. 54 .
20. ^ H. Schubert: Topology . 1975, p. 20 .
21. ↑ H.-J. Kowalsky: Topological Spaces . 1961, p. 52 .
22. ↑ G. Preuss: General Topology . 1972, p. 29 .
23. ↑ JL Kelley: General topology . 1975, p. 43 .
24. ↑ E. Harzheim, H. Ratschek: Introduction to General Topology . 1978, p. 23 .
25. ^ K. Kuratowski: Topology. Volume I . 1966, p. 43 .
26. ↑ JL Kelley: General topology . 1975, p. 43 .
27. ^ R. Vaidyanathaswamy: Set Topology . 1964, p. 57 .
28. ^ H. Schubert: Topology . 1975, p. 15 .
29. ↑ H. Herrlich: Topology I: Topological Spaces . 1986, p. 18 .
30. ↑ H.-J. Kowalsky: Topological Spaces . 1961, p. 53 .
31. ^ R. Vaidyanathaswamy: Set Topology . 1964, p. 57 .
32. ↑ JL Kelley: General topology . 1975, p. 45 .
33. ↑ J. Nagata: Modern General Topology . 1985, p. 34 .
34. ^ R. Vaidyanathaswamy: Set Topology . 1964, p. 57-58 .
35. ^ R. Vaidyanathaswamy: Set Topology . 1964, p. 58 .
36. ↑ W. Rinow: Textbook of Topology . 1975, p. 68-69 .
37. ^ K. Kuratowski: Topology. Volume I . 1966, p. 75 .
38. ↑ Cf. Rinow, p. 68. According to Hausdorff ( Grundzüge der Setlehre , p. 220), the concept goes back to Georg Cantor . In view of the possible confusion with the derivative of functions in the calculus in the topology of the designation is Derived opposite derivative preferable.
39. ^ K. Kuratowski: Topology. Volume I . 1966, p. 75 .
40. ↑ H.-J. Kowalsky: Topological Spaces . 1961, p. 53 .
41. ↑ H.-J. Kowalsky: Topological Spaces . 1961, p. 53 .
42. ↑ E. Harzheim, H. Ratschek: Introduction to General Topology . 1978, p. 2 .
43. ↑ L. Guide: General topology with applications . 1977, p. 14 .
44. ↑ B. v. Querenburg: Set theoretical topology . 2001, p. 20 .
45. ↑ The above system of axioms differs from that which Hausdorff provides in the basics (p. 213). In particular, Hausdorff always takes the validity of the axiom of separation named after him as given, which does not correspond to the modern version of the surrounding axioms .
46. ↑ J. Nagata: Modern General Topology . 1985, p. 32 .
47. ↑ G. Preuss: General Topology . 1972, p. 24 .
48. ^ H. Schubert: Topology . 1975, p. 13 .
49. ↑ L. Guide: General topology with applications . 1977, p. 14 .