What is the difference between bounded and closed

To find the factors, consider the numbers that yield a product of If we think about it, we can list all of the numbers that 24 is divisible by: 1, 2, 3, 4, 6, 8, 12, and This is a complete list of the factors of Prime factorization example: This factor tree shows the factorization of It shows that is the product of five 2s and three 3s. Prime factorization is a particular type of factorization that breaks a number of interest into prime numbers that when multiplied back together produce the original number.

Such prime numbers are called prime factors. For example, consider the number 6. Also note that 2 and 3 are prime numbers, because each is divisible by only 1 and itself.

Therefore, 2 and 3 are prime factors of 6. Now, consider the number However, 6 is not a prime factor. In this case, we must reduce 6 to its prime factors as well.

We have now found factors for 12 that are all prime numbers. Every positive integer greater than 1 has a distinct prime factorization. To factor larger numbers, it can be helpful to draw a factor tree.

In a factor tree, the number of interest is written at the top. Then, two factors of that number are found and connected below that number with branches. This process repeats for each subsequent factor of the original number until all the factors at the bottoms of the branches are prime.

Privacy Policy. Skip to main content. Numbers and Operations. Search for:. Properties of Real Numbers. Learning Objectives Use interval notation to represent sets of numbers. Key Takeaways Key Points A real interval is a set of real numbers with the property that any number that lies between two numbers included in the set is also included in the set.

To indicate that an endpoint of a set is not included in the set, the square bracket enclosing the endpoint can be replaced with a parenthesis. An open interval does not include its endpoints, and is enclosed in parentheses. A closed interval includes its endpoints, and is enclosed in square brackets. An interval is considered bounded if both endpoints are real numbers. An interval is unbounded if both endpoints are not real numbers.

Replacing an endpoint with positive or negative infinity—e. Key Terms interval : A distance in space. Learning Objectives Define the absolute value of a number. Key Takeaways Key Points The absolute value of a real number may be thought of as its distance from zero along the real number line. Learning Objectives Use set notation to represent sets of numbers and describe the properties of commonly used sets of numbers.

Key Takeaways Key Points A set is a collection of distinct objects and is considered an object in its own right. The common categories of number sets are natural numbers, real numbers, integers, rational numbers, imaginary numbers, and complex numbers. Key Terms superset : A set that contains another set. Also, the limit lies in the same set as the elements of the sequence, if the set is closed.

Then when metric spaces are introduced, there is a similar theorem about convergent subsequences, but for compact sets. At this point things get a bit abstract. So, can somebody explain the difference between compact , bounded and closed sets with examples? Part of the problem is that boundedness is a nearly useless property by itself in the context of metric spaces.

Whether a sequence in a metric space is convergent or not depends only on the open sets of the space. But is it necessarily the case that every sequence in an arbitrary metric space must have a Cauchy subsequence?

Obviously not. Nice and bounded w. Instead we have come up with the notion of total boundedness:. It does so due to the following proposition:. Since total boundedness implies boundedness, it becomes very reasonable to ask questions about the converse, e.

Or more schematically:. And again we wonder about the converse. Does Cauchy imply convergent? If it did, we would be very happy. Because then whenever we have a totally bounded space and a sequence in it, we would know that it has a Cauchy and therefore convergent subsequence.

But we are not so fortunate that this holds in arbitrary metric spaces. Which it sort of does, but it's not in the space under consideration. However, the objection does illustrate that it's not an ideal example which is why I'd like to consider instead the following:. It may not be immediately obvious that this sequence is Cauchy w. But it's far more interesting to ask what it converges to. In some sense it converges pointwise to. So what went wrong? We even have a candidate for the limit, but it doesn't fit into our space.

The sad news is that there isn't a whole lot we can do. Some metric spaces have the property that Cauchy sequences converge and those are nice and we call them complete.

The word seems to suggest that "all the limits of Cauchy sequences that should be there, are there" which is not an entirely wrong picture. And a further theorem tells is that a subset of a complete space is itself complete if and only if it's closed in the larger space. Tying it all together, we have total boundedness and completeness. As you might imagine a totally bounded complete space is a wonderful place to do analysis. Whenever you're given a sequence you know that it has a Cauchy subsequence and by completeness you know that said subsequence must be convergent.

Absolutely fantastic! But how does that tie in to the Heine—Borel "every open cover has a finite subcover" definition? The first thing we note is that a metric space which has the Heine—Borel property must be totally bounded. It is an open cover and therefore has a finite subcover.

Which is equivalent to the Heine—Borel property. But this tells us something about completeness because it easily implies the Nested Set Property from the previous section. This is all very standard material in courses on metric spaces.