On the topic of the ownership of Weathertop (or Amon Sul), Appendix A says:
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Both Rhudaur and Cardolan desired to possess Amon Sul (Weathertop), which stood on the borders of their realms...
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and then
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Arthedain and Cardolan held in force a frontier along the Weather Hills, the Great Road and the lower Hoarwell
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so the suggestion seems to be that at one point Cardolan and Arthedain held this area. And then
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A great host came out of Angmar in 1409, and crossing the river entered Cardolan and surrounded Weathertop.
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Now it could be that these were simultaneous activities (not related), but one could read this as indicating that Weathertop was part of Cardolan at this point in time.
Shortly thereafter,
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Cardolan was ravaged...A remnant of the the faithful among the dunedain of Cardolan also held out in Tyrn Gorthad (the Barrowdowns), or took refuge in the Forest behind.
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so I agree there is no clear statement here that a city was involved at the site of Tyrn Gorthad.
I guess what had me thinking that a city was close by was the statement by Bombadil when he found the brooch set with blue stones
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"Here is a pretty toy for Tom and for his lady! Fair was she who long ago wore this on her shoulder..."
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which suggests to me that Bombadil knew the princess in question. If this was just a group of warriors on the Barrowdowns, then I don't see a princess being there. Of course, it could be that Bombadil somehow knew this princess by other means, but the fact that he knew her suggested that the royal palace might be nearby. Perhaps even more circumstantial is the statement in the Tale of Years
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1409 The Witch King of Angmar invades Arnor. King Arvaleg I is slain. Fornost and Tyrn Gorthad are defended,
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which suggests to me that Fornost and Tyrn Gorthad are more or less equivalent (i.e., capitals of their respective kingdoms)...