Polytopological Space

In general topology, a polytopological space consists of a set $X$ together with a family $\_$ of topologies on $X$ that is linearly ordered by the inclusion relation ($I$ is an arbitrary index set). It is usually assumed that the topologies are in non-decreasing order, but some authors prefer to put the associated closure operators $\_$ in non-decreasing order (operators $k_i$ and $k_j$ satisfy $k_i\leq k_j$ if and only if $k_iA\subseteq k_jA$ for all $A\subseteq X$), in which case the topologies have to be non-increasing.

Polytopological spaces were introduced in 2008 by the philosopher Thomas Icard for the purpose of defining a topological model of Japaridze's polymodal logic (GLP). They subsequently became an object of study in their own right, specifically in connection with Kuratowski's closure-complement problem.

Definition

An $L$-topological space $(X,\tau)$
is a set $X$ together with a monotone map $\tau:L\to$ Top$(X)$ where $(L,\leq)$ is a partially ordered set and Top$(X)$ is the set of all possible topologies on $X,$ ordered by inclusion. When the partial order $\leq$ is a linear order, then $(X,\tau)$ is called a polytopological space. Taking $L$ to be the ordinal number $n=\,$ an $n$-topological space $(X,\tau_0,\dots,\tau_)$ can be thought of as a set $X$ together with $n$ topologies $\tau_0\subseteq\dots\subseteq\tau_$ on it (or $\tau_0\supseteq\dots\supseteq\tau_,$ depending on preference). More generally, a multitopological space $(X,\tau)$ is a set $X$ together with an arbitrary family $\tau$ of topologies on $X.$

See also

* Bitopological space

References

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Topology