Roar Stovner
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 Jun7 awarded Critic May31 awarded Scholar May31 accepted Is there a name for a collection of open sets where arbitrary intersections are open? May30 comment Is there a name for a collection of open sets where arbitrary intersections are open? Excellent! On the odd chance that somebody comes along with a better answer, I'll wait a little before accepting. May30 awarded Student May30 asked Is there a name for a collection of open sets where arbitrary intersections are open? May20 awarded Editor May20 revised homotopy direct limits Included pointer to the discussion in the comments. May20 comment homotopy direct limits I believe you are correct. Your supposition that $X_\Sigma$ is a h-direct limit of the $U_i$ is at least valid. The space $X_\Sigma$ is in fact the direct limit of the $U_i$ and since all the inclusions $U_i \hookrightarrow U_{i+1}$ are cofibrations the direct limit and h-direct limit are the same. I will edit my answer and point the reader to your argument. May19 comment homotopy direct limits Yes, that's exactly the map I had in mind, Tim! This answer to another question is related to our situation, but there the existence of a homotopy inverse is guaranteed by an algebraic argument. May19 answered homotopy direct limits Jun11 comment How to study math to really understand it and have a healthy lifestyle with free time? This really struck a nerve in the community here; no other questions produces 6 lengthy answers in 6 hours! So @Leon: Cherish the fact that you're not alone in this situation until you find a remedy. :) Jan28 awarded Supporter Nov10 answered How many ways can I make six moves on a Rubik's cube? Oct17 comment Book about technical and academic writing reddit.com/r/math/comments/9wdzo/… If you google the title of the book you will find one version, but the typesetting is so ugly you wouldn't want to read it! Oct17 awarded Teacher Oct17 answered Book about technical and academic writing