Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Calculus and Beyond Homework Help
Path-connectedness for finite topological spaces
Reply to thread
Message
[QUOTE="Wendel, post: 5763752, member: 623524"] [h2]Homework Statement [/h2] I'm trying to understand the intuition behind path-connectedness and simple-connectedness in finite topological spaces. Is there a general methodology or algorithm for finding out whether a given finite topological space is path-connected? [h2]Homework Equations[/h2] how can I determine which of the following topologies are path-connected? for the three-element set: 1. {∅,{a,b,c}} 2. {∅,{c},{a,b,c}} 3. {∅,{a,b},{a,b,c}} 4. {∅,{c},{a,b},{a,b,c}} 5. {∅,{c},{b,c},{a,b,c}} 6. {∅,{c},{a,c},{b,c},{a,b,c}} 7. {∅,{a},{b},{a,b},{a,b,c}} 8. {∅,{b},{c},{a,b},{b,c},{a,b,c}} I won't list them out, but for the four-point set there are 33 inequivalent topologies. One of which is the pseudocircle X={a,b,c,d} which has the topology {{a,b,c,d},{a,b,c},{a,b,d},{a,b},{a},{b},∅}.[h2]The Attempt at a Solution[/h2] The discrete topology is totally disconnected. Any map from the unit interval to the indiscrete topology is continuous, so it must be path-connected. Furthermore the particular point topology is path-connected. The pseudocircle is clearly path-connected since the continuous image of a path-connected space is path-connected. Furthermore it is not simply connected. From Wikipedia, connectedness and path-connectedness are the same for finite topological spaces. Thank you, apologies for the long post.[/B] [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Calculus and Beyond Homework Help
Path-connectedness for finite topological spaces
Back
Top