question archive Write a complete set of proofs for Theorem 4

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Write a complete set of proofs for Theorem 4.5 in the Croom book.

- State the hypotheses

- State the conclusions

- Clearly and precisely prove the conclusions from the hypotheses

- Results presented earlier in the text may be used and must be clearly documented

A few notes about format: use Microsoft Word; use Equation Editor for all mathematical symbols, e.g. *x* ∈ *X* or *Cl(A)* ? *Cl(X-A)*; and use the fonts Cambria and Cambria Math in size 11 so your typed work is the same font as your equations.

Theorem 4.5: Let Abe a subset of a topological space X. (1)bdy . (2)bdy A, int A, and int (X\A) are pairwise disjoint sets whose union is X. (3)bdy A is a closed set. (4) = int A ∪ bdy A. (5)A is open if and only if bdy A ⊂ (X\A). (6)A is closed if and only if bdy A ⊂ A. (7)A is open and closed if and only if bdy A = Ø. Proof: Properties (1) through (4) follow immediately from the definitions. To prove (5), note that if A is open, then A = int A by Theorem 4.3, part (2). Since int A and bdy A are disjoint by (2), then A and bdy A are disjoint, so bdy A must be a subset of X\A. For the reverse implication, suppose bdy A ⊂ X\A. Then no point of A is a boundary point of A, so every point of A is an interior point. Thus A = int A, so A is open. Statement (6) follows from the duality between open sets and closed sets: A is closed if and only if X\A is open. By (5), this is equivalent to saying that or Statement (7) is proved by combining (5) and (6): A is both open and closed if and only if bdy A is contained in both A and X\A. Since A and X\A are disjoint, this occurs if and only if bdy A = Ø According to Theorem 4.5, the points of a subset A of a space X may be of two types, interior points and boundary points. The set A may have additional boundary points outside A, however; the union of all interior points and boundary points of A is . The points of X are of three non-overlapping types: (1) interior points of A, (2) interior points of X\A, and (3) boundary points of A, which are identical with the boundary points of X\A. (Of course, any of these three sets may be empty.) The following examples are an attempt to spare the reader some of the common misconceptions about boundaries and closures in metric spaces.

MLA 8th Edition (Modern Language Assoc.)

Croom, Fred H. Principles of Topology. Dover Publications, 2016.

APA 7th Edition (American Psychological Assoc.)

Croom, F. H. (2016). Principles of Topology. Dover Publications.