Why is (0,1) not compact in topology?

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In summary: The set of all intervals (1/n,1) DOES cover the interval (0,1). If x is in (0,1) then necessarily there exists N such that 1/N < x and hence x is in (1/N,1). So taking the union of all the intervals (1/n,1) for each n gives us the interval (0,1) as required. And trivially there is no finite subcover, since if there was there would be a list (1/n_1, 1), (1/n_2, 1),... and we can pick the largest nk and the union of these intervals would just be the interval (1/n
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Here is an easier way to see why (0,1) is not compact (more intuitively, anyway)- topologically, we cannot tell it apart from the real numbers. Sure, to us looking from outside embedded in the real line it "looks small" and "non-infinite" but we could relabel the points so that it wasn't. We could take a homeomorphism from (0,1) (actually, (-1,1)) into the real line by sending x to, say tan(\pi.x).

I'm sure that you see intuitively why the real line is not compact? Think of any argument for this that you want- you will be able to directly translate it to the space (0,1) (we think of (0,1) as being without boundary on both ends).

Try thinking about the sphere with the north pole removed. You should have seen stereographic projection which defines a homeomorphism from the sphere with a point removed to the plane. So the sphere with a point removed is non-compact.

Effectively, what I'm trying to tell you is that when looking purely from a topological point of view, there are good reasons why the spaces you are looking at should be considered as non compact.
 
<h2>1. Why is (0,1) not compact in topology?</h2><p>In topology, compactness is a property that describes the behavior of a set under certain types of transformations. A set is considered compact if it is closed and bounded, meaning that it contains all of its limit points and is contained within a finite distance. The set (0,1) is not compact because it is not closed, as it does not contain its limit points of 0 and 1.</p><h2>2. How does the definition of compactness apply to (0,1)?</h2><p>The definition of compactness states that a set must be both closed and bounded in order to be considered compact. In the case of (0,1), the set is bounded as it is contained within a finite distance, but it is not closed as it does not contain its limit points. Therefore, it does not meet the criteria for compactness.</p><h2>3. Can (0,1) be made compact by changing the topology?</h2><p>No, the set (0,1) cannot be made compact by changing the topology. The topology of a set is determined by its open sets, and the open sets of (0,1) are already defined as the intervals (a,b) where 0 &lt; a &lt; b &lt; 1. No matter how the topology is changed, the set will still not be closed and therefore not compact.</p><h2>4. What is the significance of (0,1) not being compact in topology?</h2><p>The fact that (0,1) is not compact in topology has important implications in mathematics, particularly in the study of analysis and functional analysis. It is used as a counterexample to certain theorems and can help to illustrate the limitations of certain mathematical concepts.</p><h2>5. Are there other sets that are not compact in topology?</h2><p>Yes, there are many other sets that are not compact in topology. In fact, the majority of sets are not compact. Some examples include the set of real numbers (which is unbounded), the set of rational numbers (which is not closed), and the set of integers (which is not bounded).</p>

Related to Why is (0,1) not compact in topology?

1. Why is (0,1) not compact in topology?

In topology, compactness is a property that describes the behavior of a set under certain types of transformations. A set is considered compact if it is closed and bounded, meaning that it contains all of its limit points and is contained within a finite distance. The set (0,1) is not compact because it is not closed, as it does not contain its limit points of 0 and 1.

2. How does the definition of compactness apply to (0,1)?

The definition of compactness states that a set must be both closed and bounded in order to be considered compact. In the case of (0,1), the set is bounded as it is contained within a finite distance, but it is not closed as it does not contain its limit points. Therefore, it does not meet the criteria for compactness.

3. Can (0,1) be made compact by changing the topology?

No, the set (0,1) cannot be made compact by changing the topology. The topology of a set is determined by its open sets, and the open sets of (0,1) are already defined as the intervals (a,b) where 0 < a < b < 1. No matter how the topology is changed, the set will still not be closed and therefore not compact.

4. What is the significance of (0,1) not being compact in topology?

The fact that (0,1) is not compact in topology has important implications in mathematics, particularly in the study of analysis and functional analysis. It is used as a counterexample to certain theorems and can help to illustrate the limitations of certain mathematical concepts.

5. Are there other sets that are not compact in topology?

Yes, there are many other sets that are not compact in topology. In fact, the majority of sets are not compact. Some examples include the set of real numbers (which is unbounded), the set of rational numbers (which is not closed), and the set of integers (which is not bounded).

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