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I'm pretty sure it was less than 50 years ago that what are now called compact topological spaces were known as "bicompact", and "compact" meant what is now called "sequentially compact".


2

The statement that elliptic curves are modular was (I think) universally known as the Weil Conjecture back in 1970 or so when I first learned about it. Nowadays, it's generally known as the Shimura-Taniyama-Weil Conjecture (or, the Modularity Theorem). I don't know enough about the history to discuss the conditions responsible for the success of the proposed ...


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A few examples: Regarding Grothendieck's theory of schemes: These were called preschemes at first, while the word 'scheme' was reserved for separated (pre)schemes. ( Preschemes and schemes). An example from (combinatorial) topology: Nowadays the terminology simplicial set is universally used for a bunch of sets indexed over the nonnegative numbers ...



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