Connectivity spaces
The concept of connectedness -- highly used in any matter related to topology -- has been treated, in a rigorous process of axiomatization, by many researchers, with different types of approaches altogether. In this paper, an attempt is made to have a glimpse of some of them; this `review-like paper' attempts to point out the underlying resemblance amongst such approaches which have been chosen by the authors. One way to obtain connectedness via axioms has been the axiomatization of connectedness, the other way being the delineation of axiom systems describing separation. The first approach is to define a connectivity space as a pair \((X,\mathcal C)\) with \(\mathcal C\subseteq 2^X\) such that (C 0): \(\emptyset\in\mathcal C\) and (C 1): \(Z_i\in\mathcal C\) for all \(i\in I\) and \(\cap Z_i\neq\emptyset\) implies \(\cup_iZ_i\in\mathcal C\). Consequently, the concept of connected component is defined as follows: for every \(A \subseteq X\) and every \(x\in X\), the set \(A[x]=\cup\{a'\subseteq X:A'\subseteq A\), \(x\in A'\) and \(A'\in\mathcal C\}\) is called the connected component of \(x\in A\); however, a basic difference with the same concept in topological spaces is that \(\{A[x]:x\in X\}\) does not define a partition of \(A\) but only a partial partition with \(\mathcal C\subseteq 2^X\) an arbitrary collection of connected sets. A pair \((A,B)\) is termed as \(\mathcal O\)-separating, if for every connected subset \(Z\in\mathcal C\) with \(Z\subseteq A\cup B\), it holds that \(Z\cap A=\emptyset\) or \(Z\cap B=\emptyset\); the collection of all separations with respect to a given collection of connected sets \(\mathcal C\) is denoted by \(\mathfrak S_{\mathcal C}\) or simply by \(\mathfrak S\); the properties satisfied by \(\mathfrak S_{\mathcal C}\) have been delineated in terms of (S 0)--(S 2) and (SR 0)--(SR 2). In another approach to connectivity, the concept of `connectivity' opening is defined as a map \(Y\): \(X\times 2^X\to 2^X:(x,A)\to a[x]\) which satisfies certain axioms termed as (O 1)--(O 4), for all \(x\in X\) and all \(A,B\in 2^X\); defining \(\{A[x]:x\in X \text{ and } A\in 2^X\}\) to be the connected components when they satisfy (O 1)--(O 4), it is shown that there is a one-to-one correspondence between this class and connectivity spaces given by \(\mathcal C=\{Z\in 2^X:Z=A[x],A\subseteq X,x\in X\}\) satisfying (C 0) and (C 1). In an attempt to define separation without reference to any \(\mathcal C\) whatsoever, a set of pairs is defined as symmetric separation if (S 0)--(S 2) are satisfied. To describe effectively the existence of a structure in conformity with the idea of connected components, another axiom (SR 3) is incorporated; in this connection, it is shown that (SR 3) is equivalent to (SR 0)--(SR 2); the idea of connected components comes in (SR 3) and it is shown that, for a connectivity space \(\mathcal C\) on \(X\), the corresponding separation space \(\mathfrak S_{\mathcal C}\) consists of all pairs \((A,B)\) such that every connected component of \(A\cup B\) is contained either in \(A\) or \(B\). In the next phase, for symmetric separation \(\mathfrak S\) on \(X\), a set \(Z\subseteq X\) is called \(\mathfrak S\)-connected if \(Z\cap A=\emptyset\) or \(Z\cap B=\emptyset\), for all \((A,B)\in\mathfrak S\) such that \(Z\subseteq A\cup B\); the collection of \(\mathfrak S\)-connected sets is denoted by \(\mathcal C_{\mathfrak S}\) and it satisfies (C 0) and (C 1) with \(\mathcal D=\{\{A;B\}:A\neq\emptyset,B\neq\emptyset\}\), where \(\{A;B\}\mathcal XZ\) iff \(Z\subseteq A\cup B\) implies \(Z\cap A=\emptyset\) or \(Z\cap B=\emptyset\); a one-to-one correspondence between connectivity spaces and symmetric separations is established via this binary relation \(\mathcal X\). Consequently, after defining a point-wise symmetric isotone closure space \((X,W)\), with certain conditions imposed on \(\mathfrak S\) as above i.e. if \(\mathfrak S\) satisfies (S 1) and (S 2) and another axiom (S X), \(\mathfrak S\) is shown to be the Hausdorff-Lennes separation of the point-wise symmetric isotone closure space \((X, W)\). The properties of point-symmetric closure spaces are therefore characterized by their Hausdorff-Lennes separations. `Topological Connectivity Spaces' are discussed and it is shown that a symmetric separation \(\mathfrak S\) is generated by the topologically connected sets of a point-symmetric closure space if and only if \(\mathfrak S\) satisfies (S X), (SR 0), (SR 1), and (SR 2). The authors elucidate integral connectivity spaces. If \(\mathcal C\) is an integral connectivity space, then property (D) is equivalent to (c 4a); on the other hand if \((X,\mathcal C)\) be a finite integral connectivity space then (c 4a) and (c 4b) are equivalent. With the axiom of additivity of Wallace, defined as follows: \((A,B\cup C)\in\mathfrak S\) whenever \((A,B)\in\mathfrak S\) and \((A,C)\in\mathfrak S\), it is shown that, for a connectivity space \((X,\mathcal C)\), with \(\mathfrak S=\mathfrak S_{\mathcal C}\) its corresponding symmetric separation satisfying Wallace's axiom, the following condition holds: \(C,Z\in\mathcal C\), \(C\subseteq Z\), \(Z-C=M\cup N\) and \((M,N)\in\mathfrak S\) implies \(C\cup M\in\mathcal C\) and \(C\cup N\in\mathcal C\). It is also shown that, in terms of \(\mathfrak S\), as above, separation axioms can be naturally phrased. Reference to Harlick's axiom and Efremovic's axiom is made and their relations to the previous ones is also discussed. Finally, the concept of catenous functions between two connectivity spaces is discussed only to have an analogue of the famous theorem which states that \((X,T)\) is connected if and only if every continuous function from \(X\) to the totally disconnected space of two points is constant.
scientific article; zbMATH DE number 6529620
scientific article