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''lattice-valued topology, point-set lattice-theoretic topology, poslat topology''

A branch of mathematics encompassing any sort of topology using lattice-valued subsets. The following description is based on the standardization of this discipline undertaken in [[#References|[a9]]], especially [[#References|[a10]]], [[#References|[a11]]]; much additional information is given in the references below.

Let $X$ be a set and $( L , \leq , \otimes )$ any complete quasi-monoidal lattice (a cqml; i.e., $( L , \leq )$ is a [[Complete lattice|complete lattice]] with bottom element $\perp$ and top element $\top$, and the [[Tensor product|tensor product]] $\otimes : L \times L \rightarrow L$ is isotone in both arguments with $\top  \otimes \top  = \top $). Examples of complete quasi-monoidal lattices are:

complete lattices with $\otimes = \wedge$ (binary meet);

$L = [ 0,1 ]$ with $\otimes$ any of the $t$-norms $T _ { \operatorname { min } } ( a , b ) = a \wedge b$, $T _ { \text { prod } } ( a , b ) = a \cdot b$, or $T _ { m } ( a , b ) = ( a + b - 1 ) \vee 0$; or

$L = [ 0,1 ] \times [ 0,1 ]$ with $\otimes$ any of $T_{\text{min}} \times T_{\text{prod}}$, $T _ { \operatorname{prod} } \times T _ { m }$, etc.

$L$-subsets of $X$ comprise the $L$-powerset $L ^ { X } = \{ a : X \rightarrow L , a \ \text {a function } \}$, a complete quasi-monoidal lattice via the lifting of the structure of $L$. A subfamily $\tau \subset L ^ { X }$ is an $L$-topology on $X$, and $( X , \tau )$ is an $L$-topological space, if $\tau$ is closed under $\otimes$ and arbitrary $\vee$ and contains the constant mapping $\underline{ \top } $. A function ${\cal T} : L ^ { X } \rightarrow L$ is an $L$-fuzzy topology on $X$, and $( X , \mathcal{T} )$ is an $L$-fuzzy topological space, if $\mathcal{T}$ satisfies (reading $\forall$ as  "for each" ):

1) $\forall$ set $J$, $\forall \{ u_j : j \in J \} \subset L ^ { X }$,

\begin{equation*} \bigwedge _ { j \in J } {\cal T} ( u _ { j } ) \leq {\cal T} \left( \underset { j \in J } { \vee } u _ { j } \right). \end{equation*}

2) $\forall$ two-element set $J$, $\forall \{ u_j : j \in J \} \subset L ^ { X }$,

\begin{equation*} \bigotimes _ { j \in J } \mathcal{T} ( u _ { j } ) \leq \mathcal{T} \left( \bigotimes _ { j \in J } u _ { j } \right) . \end{equation*}

3) $\mathcal{T} ( \underline { \top  } ) = \top $.

The member $\mathcal{T} ( u )$ of $L$ is interpreted as the  "degree of openness"  of $u$.

Important examples of $L$-topological and $L$-fuzzy topological spaces can be found in [[#References|[a5]]], Chap. 11; [[#References|[a6]]], Kubiak's paper; [[#References|[a8]]]; [[#References|[a9]]], Chaps. 6, 8, 10; [[#References|[a9]]], Chap. 7, Sect. 2.15–2.16; [[#References|[a10]]], Sect. 7; [[#References|[a11]]], Sect. 7.

For a complete quasi-monoidal lattice $L$ and function $f : X \rightarrow Y$, one defines the powerset operators $f _ { L } ^ { \rightarrow } : L ^ { X } \rightarrow L ^ { Y }$ (the image operator) and $f _ { L } ^ { \leftarrow } : L ^ { Y } \rightarrow L ^ { X }$ (the pre-image operator) by

\begin{equation*} f _ { L } ^ { \rightarrow } ( a ) ( y ) = \vee \{ a ( x ) : f ( x ) = y \}, \end{equation*}

\begin{equation*} f _ { L } ^ { \leftarrow } ( b ) = b \circ f. \end{equation*}

It is well-known that $f _ { L } ^ {\rightarrow} \dashv  f _ { L } ^ { \leftarrow }$ and that these operators generalize the traditional operators $f^\rightarrow$ and $f ^ { \leftarrow }$. Given $L$-topological spaces $( X , \tau )$ and $( Y , \sigma )$, a mapping $f : X \rightarrow Y$ is $L$-continuous from $( X , \tau )$ to $( Y , \sigma )$ if $f | _ { \sigma } ^ { \leftarrow } : \tau \leftarrow \sigma$; and given $L$-fuzzy topological spaces $( X , \mathcal{T} )$ and $( Y , \mathcal{S} )$, a mapping $f : X \rightarrow Y$ is $L$-fuzzy continuous from $( X , \mathcal{T} )$ to $( Y , \mathcal{S} )$ if ${\cal T} \circ f ^ { \leftarrow } \geq \cal S$ on $L ^ { Y }$. The [[Category|category]] $L$-$\mathbf{TOP}$ comprises $L$-topological spaces, $L$-continuous mappings, and the composition and identities from the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f13029097.png"/> (cf. also [[Sets, category of|Sets, category of]]); and the category $L$-$\textbf{FTOP}$ comprises $L$-fuzzy topological spaces, $L$-fuzzy continuous mappings, and the composition and identities from the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290104.png"/>. It is a theorem that for all complete quasi-monoidal lattices $L$, the categories $L$-$\mathbf{TOP}$ and $L$-$\textbf{FTOP}$ are topological categories over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290110.png"/>, in the sense of [[#References|[a3]]] and [[#References|[a11]]], Sect. 1, and hence topological constructs.

The above briefly describes  "fixed-basis topology"  — topology where the complete quasi-monoidal lattice $L$, viewed as the lattice-theoretic base of powersets $L ^ { X }$ and spaces $( X , \tau )$ or $( X , \mathcal{T} )$, is fixed relative to the spaces and mappings of the category $L$-$\mathbf{TOP}$ or $L$-$\textbf{FTOP}$.  "Variable-basis topology"  permits the base to change within a category, so that each space has its own lattice-theoretic base.

To outline variable-basis topology, note that all complete quasi-monoidal lattices form a category, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290120.png"/>, in which morphisms are mappings preserving $\otimes$, arbitrary $\vee$, and $\top$; and also note that $\mathbf{FRM}$ and $\textbf{SFRM}$ embed into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290126.png"/>. One then considers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290127.png"/>, with objects the same as those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290128.png"/>, now called localic quasi-monoidal lattices, but morphisms reversed from those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290129.png"/>; and one notes that $\mathbf{LOC}$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290131.png"/> embed into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290132.png"/>.

Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290133.png"/>. The category $\mathbf{C}$-$\mathbf{TOP}$ for variable-basis topology and the category $\mathbf{C}$-$\textbf{FTOP}$ for variable-basis fuzzy topology are both  "concrete"  categories over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290138.png"/> as a  "ground"  or  "base"  category. For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290139.png"/> morphism $( f , \phi ) : ( X , L ) \rightarrow ( Y , M )$, the pre-image operator $( f , \phi ) ^ { \leftarrow } : L ^ { X } \leftarrow M ^ { Y }$ is defined by $( f , \phi ) ^ { \leftarrow } ( b ) = \phi ^ { \text{op} } \circ b \circ f$. An image operator $( f , \phi ) ^ { \rightarrow }$ is also available which, if $\phi ^ { \operatorname{op} }$ preserves arbitrary $\wedge$, satisfies $( f , \phi )^{ \rightarrow} \dashv ( f , \phi )^{ \leftarrow}$; and if $L = M , \phi ^ { \operatorname{op} } = \operatorname{id} _ { L }$, these operators reduce to their fixed-basis counterparts.

Data for the category $\mathbf{C}$-$\mathbf{TOP}$ include:

objects are topological spaces $( X , L , \tau )$ (cf. also [[Topological space|Topological space]]), where $( X , L ) \in | \mathbf{SET} \times \mathbf{C}|$ and $( X , \tau ) \in | L \square \mathbf{TOP} |$;

morphisms are continuous mappings

\begin{equation*} ( f , \phi ) : ( X , L , \tau ) \rightarrow ( Y , M , \sigma ) \end{equation*}

(cf. also [[Continuous function|Continuous function]]), where $( f , \phi ) : ( X , L ) \rightarrow ( Y , M )$ is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290155.png"/> and $( f , \phi ) ^ { \leftarrow } | _ { \sigma } : \tau \leftarrow \sigma$.

Data for the category $\mathbf{C}$-$\textbf{FTOP}$ include:

objects are fuzzy topological spaces $( X , L , \mathcal{T} )$, where $( X , L ) \in | \mathbf{SET} \times \mathbf{C}|$ and $( X , \mathcal{T} ) \in | L \square \mathbf{FTOP}|$;

morphisms are fuzzy continuous mappings $( f , \phi ) : ( X , L , \mathcal{T} ) \rightarrow ( Y , M , \mathcal{S} )$, where $( f , \phi ) : ( X , L ) \rightarrow ( Y , M )$ is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290164.png"/> and $\mathcal{T} \circ ( f , \phi ) ^ { \leftarrow } \geq \phi ^ { \operatorname{op} } \circ \mathcal{S}$ on $M ^ { Y }$.

In both categories, compositions and identities are those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130290/f130290167.png"/>.

It is a theorem that for all $\mathbf{C}\subset\mathbf{LOQML}$, $\mathbf{C}$-$\mathbf{TOP}$ and $\mathbf{C}$-$\textbf{FTOP}$ are topological over the ground $\mathbf{SET} \times \mathbf{C}$ in the sense of [[#References|[a3]]] and [[#References|[a11]]], Sect. 1. Further, these frameworks unify all the fixed-basis categories for topology given above and hence unify all important examples (referenced above) over different lattice-theoretic bases (e.g. two fuzzy real lines $\mathbf{R} ( L )$ and $\mathbf{R} ( M )$). Moreover, all purely lattice-theoretic or point-free approaches to topology — locales, topological molecular lattices, uniform lattices, etc. (see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a8]]]) — categorically embed into $\mathbf{C}$-$\mathbf{TOP}$ or $\mathbf{C}$-$\textbf{FTOP}$ (for appropriate $\mathbf{C}$) as subcategories of singleton spaces; e.g. $\mathbf{LOC}$ embeds as singleton spaces into $\mathbf{LOC}$-$\mathbf{TOP}$. Thus, the variable-basis approach categorically unifies topology and fuzzy topology as a discipline.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  P.T. Johnstone,   "Stone spaces" , Cambridge Univ. Press  (1982)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  G.-J. Wang,   "Theory of L-fuzzy topological spaces" , Shanxi Normal Univ. Publ. House  (1988)  (In Chinese)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Adamek,   H. Herrlich,   G.E. Strecker,   "Abstract and concrete categories" , Wiley  (1990)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  U. (ed.) Höhle,   "Mathematical aspects of fuzzy set theory"  ''Fuzzy Sets and Syst.'' , '''40''' :  2  (1991)  (Special Memorial Volume–Second Issue)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  "Applications of category theory to fuzzy subsets"  S.E. Rodabaugh (ed.)  E.P. Klement (ed.)  U. Höhle (ed.) , Kluwer Acad. Publ.  (1992)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  U. Höhle,   S.E. Rodabaugh,   A. (eds.) Šostak,   "Special issue on fuzzy topology"  ''Fuzzy Sets and Syst.'' , '''73''' :  1  (1995)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  W. (ed.) Kotzé,   "Special issue"  ''Quaestiones Math.'' , '''20''' :  3  (1997)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  Liu Ying–Ming,   Luo Mao–Kang,   "Fuzzy topology" , World Sci.  (1997)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  "Mathematics of fuzzy sets: Logic, topology, and measure theory"  U. Höhle (ed.)  S.E. Rodabaugh (ed.) , ''The Handbooks of Fuzzy Sets'' , '''3''' , Kluwer Acad. Publ.  (1999)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  U. Höhle,   A. Šostak,   "Axiomatic foundations of fixed-basis fuzzy topology"  U. Höhle (ed.)  S.E. Rodabaugh (ed.) , ''Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory'' , ''The Handbooks of Fuzzy Sets'' , '''3''' , Kluwer Acad. Publ.  (1999)  pp. 123–272</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  S.E. Rodabaugh,   "Categorical foundations of variable-basis fuzzy topology"  U. Höhle (ed.)  S.E. Rodabaugh (ed.) , ''Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory'' , ''The Handbooks of Fuzzy Sets'' , '''3''' , Kluwer Acad. Publ.  (1999)  pp. 273–388</td></tr></table>