Re: Disconnecting Paths that reconnect to themselves

I suspect that you deleted the paths between the points at the ends of the parallel section, or their otherwise are simply no paths there - indicated by:
'the part I want to remove does not have a blue outline and I cannot add points to it'

You probably need to use the Pen tool to close the paths.
When you attempt this you might get the Same Path/Endpoints error. If this happens, you'll probably have to either Object/Expand the object and/or ungroup the object before you can close the paths.

Try what I mentioned, and if you still can't get it to work you can always upload the file and I'm sure someone will fix it for you.

Vince

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