Are the ER and the nucleus attached?

Yes, the ER (endoplasmic reticulum) and the nucleus are attached through their outer membranes. Specifically, it is the rough ER that is attached to the nucleus.

The ER is a large membrane-bound organelle that is spread throughout the cytoplasm of eukaryotic cells. It is composed of one continuous, unbroken membrane that is made up of two layers.

There are two types of endoplasmic reticulum – smooth ER and rough ER. The rough ER lies immediately adjacent to the cell nucleus. The outer membrane of the rough ER is continuous with the outer membrane of the nuclear envelope. Both, the rough ER and the nuclear envelope, have numerous ribosomes attached to the outer membrane.

The perinuclear space that lies between the outer and inner membranes of the nuclear envelope is also connected with the lumen of the rough ER.