According to switchfromshapefile.org, there is no way to describe topological relations in a shapefile. I am aware that this can be done using a Geodatabase and I have heard of TopoJSON.
If there are any other alternatives which GIS Software works with them?
Answer
You can run the ArcGIS Tool Polygon_Neighbors on a Polygon Shapefile. The output is a Table that lists the neighbors to each Polygon. There is also an option to analyse Overlaps. This tool works with any license level. This ArcGIS functionality seems to contradict your statement, "there is no way to describe topological relations in a shapefile".
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