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You're absolutely right -- I apologise. Now I'm looking for infinite counterexamples. Maybe your example can be adapted to infinity (I don't see how yet, though).
That's right @AndrejBauer , but it doesn't follow that $\iota$ (which I now call $f$ after editing) must be the inclusion map, as Qiaochu Yuan and Mark Grant seem to be implying
No $\iota: A\to X$ can be any continous map, but you are right in that it has to be injective -- but it doesn't have to be the injection! Maybe my choice of letter was unfortunate, I will change this.
