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
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
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
Mathematics
Topology and Analysis
Heine-Borel Theorem in R^n .... Stromberg, Theorem 3.40 .... ....
Reply to thread
Message
[QUOTE="HallsofIvy, post: 6776066, member: 637751"] [LEFT][COLOR=#333333][FONT=Verdana]"Because S is compact it must have a finite subcover." No, that is not correct. In fact, it doesn't make sense. To talk about a "subcover" you must first have specified a "cover". Are you talking about the one given in the argument, $B_k(0)$, the collection of open balls with center at 0 and radius k= 1, 2, 3, ...? Notice that these balls are each contained in the next so that any point in the union of a finite number of them is in the largest, the largest "k". Since every point is S is some finite distance from 0 every point of S is in one of those so it is an open cover for S. Since S is compact there exist a finite [LEFT][COLOR=#333333][FONT=Verdana][B]sub[/B]cover of that cover. There are only a finite number of such "$B_k(0)$" so there is a largest "k". S is covered by that single set so the distance between any two points in S is less than 2k.[/FONT][/COLOR][/LEFT] [/FONT][/COLOR][/LEFT] [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Topology and Analysis
Heine-Borel Theorem in R^n .... Stromberg, Theorem 3.40 .... ....
Back
Top