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This is the first part of Q4 of §1.2 in Dieck's algtop. I'd like to prove this by showing that image of every open subset in the domain is open. But what exactly is the topology with which the multiplicative group $\mathbb{C}^*$ is endowed?

Sazed
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    Relevant maybe: https://math.stackexchange.com/questions/930876/a-covering-map-is-open ? – ThePuix Sep 19 '20 at 12:33
  • @ThePuix Yes, it seems so. So, all I have to do is first prove and then use the fact that the exponential map is a covering map. That's a very nice way of circumventing the specification of the topology on $\mathbb{C}^*$! – Sazed Sep 19 '20 at 12:49
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    Oh right, sorry that was your actual question. The topology on $\mathbb{C}^$ is the subspace topology indued by the inlcusion $\mathbb{C}^ \subset \mathbb{C}$. – ThePuix Sep 19 '20 at 12:50
  • Oh, of course it's the subspace topology! Thank you! – Sazed Sep 19 '20 at 12:52
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    $z \to e^z$ is analytic so open (open mapping theorem from complex analysis). In most curricula complex functions are covered way before any algebraic topology courses so Dieck probably assumes you know this already so there is no work to be done. – Henno Brandsma Sep 19 '20 at 13:09
  • @HennoBrandsma Yes, I really should have recalled this. Perhaps I ought to refresh my memory on complex analysis before going any further. – Sazed Sep 19 '20 at 13:17
  • Math assumes perfect recall: you are supposed to know everything that came before. And the topology facts from Complex analysis are quite relevent to algebraic topology (historically it is a precursor to many ideas from it, like manifolds, homotopy etc.). Cauchy's formula is probably not that relevant there (though I did need that later in in an operator theory course..) Arzela-Ascoli and compactifications (Riemann sphere) I also encountered first in complex analysis. – Henno Brandsma Sep 19 '20 at 13:29
  • @HennoBrandsma I see, thank you. – Sazed Sep 19 '20 at 13:38
  • A meaningless word like algtop is incapable of ever being a title of a text. – William Elliot Sep 19 '20 at 17:36

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