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

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.

## Needs help with similar assignment?

We are available 24x7 to deliver the best services and assignment ready within 3-4 hours? Order a custom-written, plagiarism-free paper

## Math homework help

Do not let math assignments discourage you from trying your best. One of our math experts can help you with math homework online. Our math tutors can help you with any level of algebra, calculus, or geometry. Ask your question to get the support you need 24/7.Order Over WhatsApp Place an Order Online