Sequentailly Compact and Connected

  • Thread starter Thread starter Paper Wings
  • Start date Start date
  • Tags Tags
    Compact
Paper Wings
Messages
7
Reaction score
0

Homework Statement



Which subset of R are both sequentially compact and connected?

Homework Equations





The Attempt at a Solution


The connected subsets of R are the empty set, points, and intervals.
The subsets of R that are compact are closed and bounded.

Thus, the subsets of R that are both sequentially compact and connected are closed, bounded continuous intervals. Is this correct?
 
Physics news on Phys.org
I'm convinced. But I don't think you need to say an interval is 'continuous'. If you mean 'without holes' isn't that automatic?
 
Yes, I see what you mean. Thank you.

Does anyone know how to close or edit the words "SOLVED" in threads?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

I Compact manifold cannot be represented by (single) parametric equation
Replies
5
Views
618
Exploring Open, Closed, Bounded, and Compact Sets in R with a Unique Metric
Replies
1
Views
2K
Question about Frechet compactness
Replies
5
Views
1K
Showing A Compact Interval Is Closed
Replies
3
Views
3K
Is Every Connected Metric Space Compact?
Replies
1
Views
2K
If A and B are compact, show AUB is compact.
Replies
9
Views
7K
I Is the Set of Integer Outputs of sin(x) Sequentially Compact in ℝ?
Replies
3
Views
3K
Compact image under every continuous function
Replies
10
Views
2K
Argue, why given Operators are compact or not.
Replies
2
Views
2K
Compactness of Nested Subsets in Metric Spaces
Replies
12
Views
2K

Hot Threads

Recent Insights

Back
Top