Week 7

Respond to one of the following two prompts:

i) Define a topological space X that is compact and show that it is compact using the definition of compactness and/or relevant theorems.

ii) Define a topological space that is not compact and show that it is not compact using the definition of compactness and/or relevant theorems.

In addition look at your classmates’ topological spaces. Verify their compactness or non-compactness.

For example: Every closed and bounded interval [a, b] in ℝ with the standard topology is compact.

Proof: Let Obe an open cover of [a, b]. We need to show that there is a finite subcover of [a, b]. Suppose that there is no finite subcover of [a, b].

Consider dividing the interval [a, b] in half into two half-intervals [a,a+b2] and [a+b2, b]. One of these half-intervals does not have a finite subcover from O; otherwise, [a, b] would have a finite subcover. Let [a1, b1] be the half-interval that does not have a finite subcover from O. We can repeat this process of dividing in half, and let [a2, b2] be the half-interval of [a1, b1] that is not finitely-coverable. Then, we get an infinite collection of half-intervals [an, bn] that are not finitely-coverable.

We have that for each n=1, 2, 3, …:

i) [an, bn] ⊂ [an+1, bn+1]

ii) bn-an = b−a2n

iii) [an, bn] is not finitely-coverable

By Cantor’s Nested Intervals Theorem (Theorem 2.11 of Croom), ∩∞n=1[an,bn] is nonempty. Let x be in this intersection. Then, x∈[a, b]. Since O is an open cover of [a, b], there is an open set O in O such that x∈O. Since O is open in ℝ, there is an open interval (c, d)⊂O containing x; in particular, there is an epsilon-neighborhood (x−ε, x+ε)⊂(c, d)⊂O containing x.

Let N be sufficiently large such that b−a2N < ε. Since x lies in ∩∞n=1[an,bn], x lies in [aN, bN]. Note that [aN, bN]⊂(x−ε, x+ε) because bN−aN=b−a2N < ε. So [aN, bN]⊂(x−ε, x+ε)⊂O, which implies that [aN, bN]⊂O. That is, [aN, bN] is covered by one open set in O; in other words, [aN, bN] is finitely-coverable. This contradicts the fact that all of the half-intervals [an, bn] are not finitely-coverable.
We can conclude that O has a finite subcover. Therefore, every open cover of [a, b] has a finite subcover, and [a, b] is compact.

Don't use plagiarized sources. Get Your Custom Essay on

Week 7

Just from $13/Page

The price is based on these factors:

Academic level

Number of pages

Urgency

Basic features

- Free title page and bibliography
- Unlimited revisions
- Plagiarism-free guarantee
- Money-back guarantee
- 24/7 support

On-demand options

- Writer’s samples
- Part-by-part delivery
- Overnight delivery
- Copies of used sources
- Expert Proofreading

Paper format

- 275 words per page
- 12 pt Arial/Times New Roman
- Double line spacing
- Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Delivering a high-quality product at a reasonable price is not enough anymore.

That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read moreEach paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read moreThanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read moreYour email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read moreBy sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more
## Recent Comments