Editing shapefile from QGIS in ArcMap causes errors according to QGIS topology check?

Question

I noticed Polygon shapefiles that were edited with ArcMap yield many errors in the QGIS Topology checker.

The picture below shows a polygon Shapefile. I made one error on purpose so the topology checker shows there is just one error (checking for overlap within the layer)

Next I edit this layer in ArcMap, creating a hole between two of the polygons

Then I use the QGIS topology checker again, yielding 3 additional errors.

Now my questions are:

Can I change some settings in ArcMap so that it will not create this
difference? I think it might be saving the vertices of the edited
polygon with a lower precision. 
if it is not possible to fix it in ArcMap, then what is the quickest method to clean these tiny errors quickly afterwards with either QGIS or PostGIS?

Etiënne Thomassen · Answer

The method suggested in the comments by csk works for cleaning the geometries

I created the errors in the below image using ArcMap 10.6 by simply moving vertices and placing them back on their original location. In QGIS the topology checker reports these overlaps.

Using snap geometries to layer from the processing toolbox resulted in a shapefile without overlap or gaps. See below screenshot for the settings (I think the English name for the mode will be snap to anchor nodes as suggested by csk)

Editing shapefile from QGIS in ArcMap causes errors according to QGIS topology check?

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