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June 13th, 2018, 09:19 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory  Topology from the start: genera and shapes
I see topology as very simple or very complicated. Elementary are stretchable 3d objects with a conserved number of holes (genera) in them. But topology may have the greatest number of references by PhDs in all mathematics. What have I been missing out on? Would someone walk me through the first steps when starting on a deformable body with or without holes and some simple applications in theory. You may have seen my post on singular genera, that is, when a hole (e.g., one of a torus) is shrunken to a singularity. Would the body conserve genus, lose one, or experience differentiation? 
June 14th, 2018, 12:15 AM  #2 
Senior Member Joined: Aug 2012 Posts: 2,262 Thanks: 689 
Topology is the study of properties invariant under continuous transformation. So there's a branch of topology called general topology (as opposed to algebraic topology, where they count holes, draw loops around holes, and the like). General topology is the abstract study of continuity if you think of it that way. What makes a function continuous is how you define the open sets. The idea of open and closed intervals of real numbers is generalized to various other ways to think about open sets and how they determine which functions are continuous. So there are these two basic aspects or branches of topology, general and algebraic, and they have a very different flavor, almost like two totally different subjects. Last edited by skipjack; June 14th, 2018 at 12:02 PM. 
June 14th, 2018, 04:37 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 
I now understand, in my problem stated above, that the singularity which the hole of a torus (with genus one) can shrink to transforms that body into genus zero, since a line can pass continually through a point.

June 17th, 2018, 11:16 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Topology is the study of continuity in its most general form.

June 21st, 2018, 10:50 PM  #5 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 
Please give me some simple examples of continuity's role in topology.

June 22nd, 2018, 12:13 PM  #6 
Senior Member Joined: Aug 2012 Posts: 2,262 Thanks: 689  It's difficult to respond to this question because topology is literally the study of continuity. It's like asking for an example of Catholicism's role in the affairs of the Vatican. Maybe that metaphor's a little stretched. Which is to say, it's deformed without being ripped or torn. It's deformed continuously. Which is what topology is all about.

June 24th, 2018, 01:16 PM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 
How about some simple and varied applications of topology, like The Seven Bridges of Königsberg?

June 25th, 2018, 01:02 AM  #8 
Senior Member Joined: Oct 2009 Posts: 772 Thanks: 279  

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genera, shapes, start, topology 
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