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2 Boundedness of Subsets of Rd 2 Boundedness of subsets of Rd In this module we will meet, among others, the following terms: bounded, boundary, closed. These terms each have their own precise definitions in this module. It is essential that you are confident with these definitions and can work with them. In particular, you should be able to investigate whether or not particular examples of subsets of Rd are closed or bounded, and to determine the boundaries of such sets. Warning! For those of you taking the module G12VEC, there is an unavoidable conflict in terminology. The standard terminology in these two rather different areas of mathematics is long-established. The term ‘bounded’ will have the same meaning in both modules, but the G12VEC notions of a ‘closed surface’ and a ‘surface with boundary’ involve different usages of the words ‘closed’ and ‘boundary’. You should ensure that you work with the appropriate definitions in each module. 1 We now introduce some standard subsets of Rd. 2.1 Open and closed balls in Rd Convention. In this module, when we say ‘Let r > 0’, we mean that r is a positive real number (and the same goes for other letters). Let r > 0 and x ∈ Rd. The (d-dimensional) open ball with radius r and centre x is the set d Br(x) = {y ∈ R | ky − xk < r} . The closed ball with radius r and centre x is the set ¯ d Br(x) = {y ∈ R | ky − xk ≤ r}. The open unit ball is the set d B1(0) = {x ∈ R | kxk < 1} and the closed unit ball is the set ¯ d B1(0) = {x ∈ R | kxk ≤ 1}. 2 For d = 2, Br(x) is the disc with radius r about x without its ‘boundary’. B¯r(x) is the same disc including its ‘boundary’. Gap to fill in 3 We will give a formal definition for the term ‘boundary’ later when we we investigate the topology of Rd. Exercise. What are the open/closed balls when d = 1? Gap to fill in 4 2.2 Bounded sets and unbounded sets In this section, we discuss bounded sets and unbounded sets. As with all concepts in this module, it is very important that you understand these definitions and examples. Informal/intuitive notions can be useful, but you will be expected to be able to work confidently and reliably with definitions and examples. The word ‘bounded’ can be particularly confusing (see the module FAQ) until you have worked with and understood some examples. Our first examples of bounded sets will be the open balls and closed balls introduced above. 5 Definition 2.2.1 A subset S ⊆ Rd is said to be bounded if there exists a real number R > 0 such that S ⊆ B¯R(0), i.e., for all x ∈ S, we have ||x|| ≤ R. A set which is not bounded is said to be unbounded. Note that R must not, of course, depend on x here! (Otherwise every set would be bounded.) Gap to fill in 6 Example 2.2.2 (1) Open or closed balls centred on the origin 0 are bounded. Gap to fill in 7 (2) All open/closed balls in Rd are bounded, whether or not they are centred on the origin. Let x ∈ Rd and let r > 0. Then B¯r(x) and Br(x) are both bounded. Gap to fill in 8 (3) The empty set ∅ is bounded, but Rd is unbounded. Gap to fill in 9 (4) Every subset of a bounded set is also bounded. Gap to fill in 10 (5) In the special case of R = R1 (i.e. when d = 1), we note that a set S ⊆ R is bounded if and only if it is both bounded above and bounded below in the sense discussed in G11ACF. Note that these concepts from G11ACF do not make sense directly in higher dimensions. Note that there is no connection at all between the terms ‘bounded’ and ‘boundary’. In particular, a bounded set need not include its boundary. 11 Some further important examples of bounded and unbounded sets are given below, when we discuss d-cells in Rd. There are many equivalent definitions of the term ‘bounded’. For example, a set S is bounded if and only if the set of all possible distances between pairs of points of S is bounded above in R. Thus non-empty bounded sets can also be described as sets which have finite diameter. See question sheets for details. Gap to fill in 12 2.3 Intervals and d-cells We begin by discussing bounded intervals in R. Let a and b be real numbers, with a ≤ b. Then we have four types of bounded interval associated with a and b. First we have the closed interval, [a, b] = {x ∈ R | a ≤ x ≤ b} ⊆ R and the open interval ]a, b[ = (a, b) = {x ∈ R | a < x < b} . We may use either of these two notations for open intervals. The new notation is useful in order to avoid confusion of the ordered pair (a, b) with the open interval ]a, b[. We also have the half-open intervals ]a, b] = (a, b] = {x ∈ R | a < x ≤ b} and [a, b[ = [a, b) = {x ∈ R | a ≤ x < b} . 13 For a = b, [a, b] is just the single-point set {a}, and the other intervals are empty in this case. Warning! You should not attempt to apply the term ‘half-open’ to a set which is not an interval. Most subsets of R are not intervals! Gap to fill in 14 Analogues of intervals in Rd can be obtained by taking cartesian products of d intervals to form d-cells. This gives us rectangles in R2, cuboids in R3 and hyper-cuboids in higher dimensions. Note. In 2006-7 (and before), d-cells were described as intervals in Rd. Let a1 ≤ b1, a2 ≤ b2, ..., ad ≤ bd be real numbers. Corresponding to these we have closed intervals [ai, bi] and open intervals ]ai, bi[ (1 ≤ i ≤ d). The corresponding closed d-cell in Rd is defined to be [a1, b1] × [a2, b2] × · · · × [ad, bd] d = {x ∈ R | a1 ≤ x1 ≤ b1, . , ad ≤ xd ≤ bd} and the corresponding open d-cell is ]a1, b1[ × ]a2, b2[ × · · · × ]ad, bd[ d = {x ∈ R | a1 < x1 < b1, . , ad < xd < bd}. 15 If d > 1 then there are many more ‘half-open’ combinations possible than just 2. For example, when d = 2 and a1 = a2 = 1, b1 = b2 = 2 we have the closed 2-cell [1, 2] × [1, 2] and the open 2-cell ]1, 2[ × ]1, 2[ Gap to fill in 16 and 14 other ‘half-open’ 2-cells such as Gap to fill in 17 In the same way, there are 64 types of bounded 3-cell in R3 and 22d types of bounded d-cell in Rd. We have not proved that the above d-cells are bounded. You may check this claim directly from the definitions. Alternatively, you can deduce it from results on the question sheets. 18 We may also form unbounded d-cells by replacing some of the ak by −∞ and/or some of the bk by ∞ (which should be thought of as meaning +∞ here). Note: ±∞ are not real numbers, so that these must be excluded from the intervals. Gap to fill in 19.
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