Show That 0 1 Is Not Compact

Understanding Compactness: Why the Open Interval (0,1) Fails the Test

In the vast landscape of mathematical analysis and topology, the concept of compactness stands as a cornerstone, distinguishing well-behaved sets from those that are not. It is a property that generalizes the familiar notion of a set being "closed and bounded" in Euclidean space, a principle famously captured by the Heine-Borel Theorem. However, this theorem applies specifically to ℝⁿ. To truly grasp the nature of compactness, we must look beyond this special case and examine its fundamental definition: a set is compact if every possible open cover has a finite subcover. This article will demonstrate, through clear reasoning and a classic counterexample, that the open interval (0,1)—despite being bounded—is not compact. Its "openness" is the critical flaw that allows us to construct an open cover with no finite subcover, revealing a deep structural difference from its closed counterpart, [0,1].

1. The Fundamental Definition: Open Covers and Finite Subcovers

Before proving anything, we must solidify our understanding of the tool we are using. The definition of compactness is often called the "open cover definition."

• An open cover of a set S is a collection of open sets, say {Uᵢ}, such that S is entirely contained within the union of all these Uᵢ. In symbols, S ⊆ ⋃ᵢ Uᵢ.
• A finite subcover is a finite sub-collection of those open sets {Uᵢ₁, Uᵢ₂, ..., Uᵢₙ} that still covers S. That is, S ⊆ Uᵢ₁ ∪ Uᵢ₂ ∪ ... ∪ Uᵢₙ.

A set S is compact if every conceivable open cover of S possesses at least one finite subcover. This is a powerful statement about the set's "finiteness" at a topological level. It means you cannot "escape" the set by covering it with infinitely many sets that each only capture a small piece; you can always select just finitely many of those sets to do the entire job.

Our goal is to show that for S = (0,1), there exists at least one open cover for which no finite subcollection can cover the entire interval. Finding this single counterexample is sufficient to prove non-compactness.

2. Crafting the Counterexample: The Shrinking Neighbourhoods

The most elegant and commonly used counterexample for (0,1) exploits the fact that the interval does not contain its endpoints, 0 and 1. We can construct an open cover that "approaches" these endpoints but never actually reaches them with any single set in a finite selection.

Consider the following infinite collection of open intervals: C = { (1/n, 1) | n ∈ ℕ, n ≥ 2 }

Let's verify this is indeed an open cover of (0,1):

1. Each set is open: (1/n, 1) is an open interval in ℝ.
2. The union covers (0,1): Take any point x in (0,1). Since x > 0, we can choose a natural number N so large that 1/N < x (by the Archimedean property of real numbers). Then, for this N, x ∈ (1/N, 1). Therefore, x is in at least one set of the collection C. Since x was arbitrary, every point in (0,1) is covered. Thus, C is an open cover.

Now, the crucial test: Can we find a finite subcover? Suppose we try to pick finitely many sets from C. Let these be: (1/n₁, 1), (1/n₂, 1), ..., (1/nₖ, 1), where n₁, n₂, ..., nₖ are some finite natural numbers (all ≥ 2).

Let N = max{n₁, n₂, ..., nₖ}. This is the largest denominator among our chosen sets. The set with the smallest left endpoint among our finite selection is (1/N, 1), because as n increases, 1/n decreases. Therefore, the union of our finite subcollection is precisely (1/N, 1).

Here is the problem: The point x = 1/(N+1) is in the original set (0,1). Is it in (1/N, 1)? No, because 1/(N+1) < 1/N. Therefore, this point is not covered by our finite union.

Conclusion: No matter which finite number of sets we pick from C, there will always be a smallest left endpoint (corresponding to the largest n we chose). Any point in (0,1) that lies between 0 and this smallest left endpoint will be left uncovered. Hence, the open cover C has no finite subcover.

Since we have found a single open cover of (0,1) that admits no finite subcover, the interval (0,1) is not compact.

3. The Intuitive "Why": The Missing Endpoints

The proof above is rigorous, but what is the intuitive picture? Imagine you are trying to cover the entire open interval (0,1) with blankets (the open sets). Our cover C uses blankets that all have one edge at 1 and the other edge at points 1/2, 1/3, 1/4, ... getting closer and closer to 0 but never touching it.

• To cover the point very close to 0, say 0.0001, you need a blanket that starts before 0.0001. That means you need the set (1/n, 1) for some very large n (where 1/n < 0.0001).
• If you try to use only finitely many blankets, you are only using finitely many such large n values. There will be a largest n you used, meaning the leftmost edge of your combined blankets is at 1/N for some finite N.
• The region (0, 1/N) remains uncovered because none of your blankets reach

This behavior is fundamentally tied to the topological boundary of the set. The interval (0,1) is bounded but not closed; it lacks its limit points 0 and 1. The open cover C exploits this deficiency by using sets whose left endpoints approach the missing point 0. Any finite selection inevitably leaves a neighborhood of 0 uncovered, precisely because 0 is not an element of the space to "anchor" the cover. In contrast, a closed interval like [0,1] is compact (by the Heine-Borel theorem), and no such open cover without a finite subcover can be constructed—the inclusion of the endpoints forces any open cover to include sets that collectively cover a neighborhood of 0 and 1, ensuring a finite subcover exists.

This example crystallizes the essence of compactness in metric spaces: it is not merely about boundedness, but about completeness in a topological sense. A set is compact if every open cover is "reducible" to a finite number of sets that still accomplish the same covering task. The failure of (0,1) demonstrates that an infinite process of "getting closer and closer" to a missing boundary point cannot be truncated finitely without losing coverage. This property has profound consequences in analysis, guaranteeing, for instance, that continuous functions on a compact set attain their maxima and minima (the Extreme Value Theorem) and that sequences have convergent subsequences (sequential compactness). The open interval (0,1) lacks these guarantees, serving as a foundational example of how the absence of boundary points can undermine global topological properties.

In conclusion, the open interval (0,1) is not compact because the specific open cover C = { (1/n, 1) | n ∈ ℕ, n ≥ 2 } admits no finite subcover. This failure stems directly from the set's openness, which allows points arbitrarily close to 0 to remain uncovered by any finite union of sets from C. The example underscores that compactness requires a set to be "complete" in a way that prevents such infinite, asymptotic escapes to a missing boundary, a condition satisfied by closed and bounded subsets of ℝⁿ but violated by open intervals like (0,1).