What noun means "the state of being intrigued/amazed by something"?
For example, the state of being poor is poverty.
Is there such a word for being intrigued/amazed?
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This may be a bit too direct, but
the state of being amazed is amazement.
The exact word which fits your purpose best will depend on the context, but in my mind amazement simply implies being surprised, shocked, or dazzled while the other words imply that as well as an intense interest to discover more about the subject. In that sense curiosity, wonder, and fascination are broader, more encompassing terms than mere amazement.
Any of them would probably be a good fit, but fascination seems to come closest to expressing the same 'dazzled' sense as amazement, while also expressing intrigue.
You might consider:
awe (n) 1. A mixed emotion of reverence, respect, dread, and wonder inspired by authority, genius, great beauty, sublimity, or might: We felt awe when contemplating the works of Bach. The observers were in awe of the destructive power of the new weapon.
May I recommend wonderment:
1 astonishment, awe, or puzzlement
2 something that produces wonder; a marvel
Note that wonderment captures both astonishment and puzzlement, a close cousin to intrigue (which M-W defines as "to arouse the interest, desire, or curiosity of (as by beguiling or baffling)").