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![1: The topological space pX, T q and semi-flow φ [left 1]. A discretization of X with Tconsistent pre-order pX , ďq [middle 2, 3]. The relation Θ ´1 [middle 4] which generates the T 0 -consistent pre-order pX , ď 0 q [right 5]. The discretization map disc : X Ñ X is continuous with respect to both topologies. Common coarsening of pX , ďq and pX , ě 0 q [right 6] resulting in a Morse pre-order pX , ď : q [right 7]. All pre-orders are represented by their Hasse diagrams.](publication/362011476/figure/fig5/AS:1177942164021259@1657854705973/The-topological-space-pX-T-q-and-semi-flow-ph-left-1-A-discretization-of-X-with.png)
1: The topological space pX, T q and semi-flow φ [left 1]. A discretization of X with Tconsistent pre-order pX , ďq [middle 2, 3]. The relation Θ ´1 [middle 4] which generates the T 0 -consistent pre-order pX , ď 0 q [right 5]. The discretization map disc : X Ñ X is continuous with respect to both topologies. Common coarsening of pX , ďq and pX , ě 0 q [right 6] resulting in a Morse pre-order pX , ď : q [right 7]. All pre-orders are represented by their Hasse diagrams.
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![Fig. 1.4: The SC-graded chain complex pC tile pXq, ∆ tile q [left] of...](publication/362011476/figure/fig3/AS:1177942164013070@1657854705935/The-SC-graded-chain-complex-pC-tile-pXq-tile-q-left-of-Fig-11-computed-using-Z_Q320.jpg)
![Fig. 5.1: The topological space pX, T q and semi-flow φ [left 1]. A...](publication/362011476/figure/fig5/AS:1177942164021259@1657854705973/The-topological-space-pX-T-q-and-semi-flow-ph-left-1-A-discretization-of-X-with_Q320.jpg)
To capture the global structure of a dynamical system we reformulate dynamics in terms of appropriately constructed topologies, which we call flow topologies; we call this process topologization. This yields a description of a semi-flow in terms of a bi-topological space, with the first topology corresponding to the (phase) space and the second to...
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