Polygons sharing boundaries

Question

This is the idea: Lets say I have two polygons laying next to each other. If i start zooming in on the boundary, a small gap will appear between them. I want the two polygon boundaries to merge into one single boundary. As a result, when I start to correct the boundary (by dragging the boundary to the left) the feature of the polygon on the right will fill up the new space, i.e; not create an open space between the two polygons.

I've looked at some other threads on the forum and found one that uses Integrate (ArcToolbox -> Data management Tools), but It didn't really do what i wanted it to.

Any help is appreciated.

ArcMap 10.0

MarcinGasior · Answer

...there is no "New->Topology" when I right-click. 
Could this be because I don't have the proper license?

Yes, Geodatabase topology requires at leas ArcEditor (Standard) license.
But instead of geodatabase topology you can create temporary map topology in edit session.
Look at this ArcGIS tutorial exercise for example: http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//001t00000021000000.htm

LarsAnker-Rasch · Answer

Learn and use topology rules/tools.

http://resources.arcgis.com/en/help/main/10.1/index.html#/An_overview_of_topology_in_ArcGIS/006200000001000000/

These are related to the align tools Alexey points out and they are available in 10.0.

Thank you. But I immediately get problems using that solution. I right-click on my feature dataset to create a Topology, but there is no "New->Topology" when I right-click. 
Could this be because I don't have the proper license?

MarkBoucher · Answer

Learn and use topology rules/tools.

http://resources.arcgis.com/en/help/main/10.1/index.html#/An_overview_of_topology_in_ArcGIS/006200000001000000/

These are related to the align tools Alexey points out and they are available in 10.0.