Wolfram Language

Explore Hierarchical Structure of Topological Space Types

Topological spaces are classified based on a hierarchy of mathematical properties they satisfy. As a result, some space types are more specific cases of more general ones. This information is encoded for "TopologicalSpaceType" entities with the "MoreGeneralClassifications" property.

For example, a Banach space is also a topological space of the following types.

The network of hierarchical relation implications can be visualized directly using the "RelationshipGraph" property, which returns the full network with the space in question highlighted.

The relationship graph is a Wolfram Language graph expression, so built-in commands can be used to create a more streamlined version in which nodes are shown as points with tooltips and directed edges are more easily visible.

As can be seen from the graph, some nodes lie at the bottom of the implication chain and hence have no more general classifications. Such leaf nodes correspond precisely to topological space types for which the "MoreGeneralClassifications" property is missing.

The same list can be obtained by identifying the nodes in the graph that have vertex out-degree of 0.

Of more interest than space types corresponding to leaf nodes are those that correspond to "central" nodes according to some centrality measure.

Picking a few standard centrality types, you can easily find the topological space types that are most central by these measures.

These results can be summarized visually by highlighting the relevant nodes in the relationship graph.

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