Review of metric spaces (2005)(en)(4s)
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It is true, however, that 0 and 1 are boundary points of 0 , 1 in X and that 7 and 8 are boundary points of 7 , 8 in X. Indeed, the reverse may be the case. There are many sets with empty boundary that have subsets with as large a boundary as possible.
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To appreciate this, consider R as a subset of itself. It has empty boundary 3. But there are many other sets that have this nice property; they are said to be closed and will be studied in more detail in Chapter 4. Here we look at a few such sets. Then F includes its own boundary in X. What is the boundary of K in R? To answer this question, we shall look separately at real numbers that are in K and at those that are not in K.
We note here that our question would be valid but worthless if K were empty; K is, in fact, a highly structured non-empty set. We leave it to the reader to prove this Q 3. Boundary Since Im is a union of intervals of R, it follows that z is not in any of them.
Since each of the intervals includes its boundary, the distance from z to each of them is not zero. An argument similar to the one given above shows that each point of the graph is a boundary point of the graph in R2 with the Euclidean metric. In this case, however, there are points not in the graph that are boundary points. But the observant reader will note that none of these extra boundary 3. We have seen in 3. This phenomenon, though of interest, is not unusual. The answer is no, and Q is a counterexample when regarded as a subset of R: its boundary is R, and R has empty boundary.
We salvage something, however, in the inclusion of 3. Boundary Theorem 3.
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We know from 3. The answer to this question is not straightforward. We shall see in 3. In particular, we 3. In order to prove ii , we make use of i and 3. This was exactly the situation we observed in 3. A quick check reveals, however, that the outcome is not always so neat. What does remain unaltered is, in the case of appending boundary points, the closure of the set and, in the case of removing boundary points, the interior of the set—as we shall see in 3. The process of appending to a set all of its boundary points yields its closure.
The concepts of interior and closure are relative ones: just as the boundary of a set depends on the metric space in which the set is considered to reside, so also do its closure and interior. The members of the interior of S are called interior points of S, and the members of the exterior of S are called exterior points of S.
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Using the information from 3. Our intuition might be that boundaries have empty interior. Such intuition would be backed up by the 3. Is the same true of all boundaries? Indeed it is not. An immediate counterexample is given by the fact that the interior of the boundary of Q in R is R itself. Proof These two results follow immediately from 3. Then iii is 46 3. Boundary obtained by replacing S by S c in ii and noting that the exterior of S c is the interior of S 3.
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Corollary 3. Then the interior, the boundary and the exterior of S are mutually disjoint and their union is equal to X. A hole-in-heart subset of a square metric space. Its boundary is shown in black, its interior is shown in dark grey, and its exterior is shown in light grey; together they make up the whole space. Indeed, we have learnt in 3. In such hopes we are not disappointed 3.
Now B. The closure of a set S in a metric space X is constructed by appending to it all of its boundary points; the interior of S is constructed by removing all of its boundary points. The intention is to produce the smallest superset of S that includes its own boundary and the largest subset of S that is disjoint from its own boundary.
Is this intention realized? In particular, does the closure of S include its own boundary, which may, as we have learnt from 3. The answer to this question is yes, as we see in 3. Later, in 4. So, by 3. Since, by 3. For ii , we have, by 3.
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So, using 3. Then iii and iv follow immediately from i and ii , respectively. Proof Using 3.
Is a union of closures the same as the closure of the union? Is an intersection of closures the same as the closure of the intersection? If we do not have equality, do we get inclusion in either case? We ask similar questions about interiors. The full answer to these questions 3. Summary This chapter opened with a discussion about boundary points of subsets of a metric space.
We then examined how boundary points relate to isolated points and accumulation points. We have talked about sets with empty boundary and how they relate to connectedness, and sets that include their boundaries closed sets. We have explored boundaries of unions and intersections of sets. We have looked at the relationships that exist between closure and interior and have examined how they behave under the basic set-theoretic operations. Let C denote the collection of constant functions in F. Some numbers have two expansions of each type, one terminating and the other ending in a recurring sequence of nines for decimal expansion or of twos for ternary expansion.
For example, the.