What's the difference between frontier and border?
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For example, which sentence is correct?
a.The frontier between France and Spain.
b.The border between France and Spain.
Is this correct?--> "The frontiers of knowledge"
a.The frontier between France and Spain.
b.The border between France and Spain.
Is this correct?--> "The frontiers of knowledge"

Frontier is one of those words that I've always understood when it was said but didn't have an exact definition of it.
Here's a definition I looked up for frontier:
1.
a. An international border.
b. The area along an international border.
2. A region just beyond or at the edge of a settled area.
3. An undeveloped area or field for discovery or research: theories on the frontier of astrophysics.
I think that a border is talking about the precise "edge" of something (like a country), and frontier can be talking about the region near that edge. According to the dictionary entry it seems that a frontier can also refer to the "precise edge" of a country.
I think that all three of your example sentences are correct.