binary star - Is the angular resolution of a telescope irrespective of used eye-piece?

The Raleigh criterion is the maximum theoretical limit that ignores the architecture, quality, and state of maintenance of optics. It basically says "assuming the optics in this instrument are PERFECT, this is the resolution you could get out of it". It's a calculation that looks only at the diameter and ignores everything else. In other words, no matter how good the instrument, you cannot beat Raleigh - but you could make things worse.

In practice, of course things are worse. Take a parabolic mirror, as used in many telescope architectures, such as the newtonian. All parabolic mirrors generate perfect images only in the center of the field of view. Anything off-center is subject to coma, an aberration that even "perfect" paraboloids will exhibit. So the real resolving power gets worse as you move towards the edge of the field.

On top of that, you have to add real-world manufacturing imperfections that any mirror will have. Also add distortions caused by temperature differences, etc.

All these contribute to distort the image formed in the focal plane of the primary mirror. The eyepiece's role is to examine and magnify that image, for you to see. That's how telescopes work - primary optics form an image in the primary focal plane, which is then examined with the eyepiece.

Of course the quality of the eyepiece will contribute to further degradation of what you actually see. Even with "magic" primary optics, if you had a "perfect" image in prime focus, a bad eyepiece will blur it. In real life, most eyepieces are at least half-decent in the center of the field (some are not), but quality degrades towards the edge. High quality eyepieces will not introduce visible degradation in the center and over most of the field. Top of the line eyepieces will not degrade the image visibly anywhere.