Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
I can not accept both nice answers(similar). So, I decided to do this: for one case, one upvote but not ✔, and for another one, ✔ but not upvote. Sorry.
I can not accept both nice answers(similar). So, I decided to do this: for one case, one upvote but not ✔, and for another one, ✔ but not upvote. Sorry.
Everywhere, compact inverse image is compact. I am taking this as a definition. Just as a note, all my surfaces are boundaryless, so there is no chance of considering properness in terms of the boundary.