"Weaker" versus "stronger" topologies
When I first learned point-set topology, the textbook said analysts and topologists have incompatible conventions about the meanings of "stronger" and "weaker" as applied to topologies. One of them means "finer" and the other "coarser," but which is which depends on whether you're an analyst or a topologist. The words "finer" and "coarser," on the other hand, are so suggestive that it's impossible to get confused about which is which.
Can someone explain why one convention may make sense in one context and the opposite convention in another? And also give us any relevant mnemonic devices?
Postscript in response to a comment: TOPOLOGY A First Course, by James R. Munkres, published in 1975, page 75:
Many mathematicians use the words "weaker" and "stronger" in this context. Unfortunately, some of them (particularly analysts) are apt to say that $\mathcal T'$ is stronger than $\mathcal T$ if $\mathcal T'\supset\mathcal T,$ while others (particularly topologists) are apt to say that $\mathcal T'$ is weaker than $\mathcal T$ in the same situation! If you run across the terms "strong topology" or "weak topology" in some book, you will have to decide from the context which inclusion is meant. We shall not use these terms in this book.
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