1080i doesn't indicate the resolution of the display.

You have to look carefully at what the set actually lists as its display resolution, regardless of what inputs it accepts. Many TVs can downscale an incoming high resolution signal. And any even without downscaling, a set only needs 540 lines of resolution to display 1080i.

A TV that advertises 1080i/720p has at most 720 lines of resolution.

EDIT: Let me rephrase that... A TV that advertises 1080i/720p has at most the capability to display an HD resolution of 720 lines (1280x720), using a display that is likely that same native resolution or perhaps 1365x768 or maybe even 1422x800. Get the actual display's native resolution to find out what you're buying and always carefully test a set beforehand to see if it meets your requirements.