Introductory topology : exercises and solutions by Mohammed Hichem Mortad

By Mohammed Hichem Mortad

The publication deals an excellent creation to topology via solved routines. it truly is regularly meant for undergraduate scholars. such a lot routines are given with particular solutions.

Readership: Undergraduate scholars and teachers in topology

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Let R be equipped with its usual topology. Let A be a non-empty and bounded subset of R. (1) Show that inf A, sup A ∈ A. (2) Does this result remain valid in other topologies on R? ◦ (3) Do we have an analogous result for A? 13. Let A be a non-void and bounded above subset of R. (1) Show that in usual R, sup A = sup A. (2) Show that the previous result need not hold in another topology of R. ◦ (3) In the usual topology of R again, do we have sup A = sup A? 34 3. 14. Let (X, d) be a metric space.

24. On R2 , consider the collection (B(0, r))r where r is a positive number allowed to be +∞ as well. (1) Show that the collection (B(0, r))r defines a topology on R2 . (2) Is this topology Hausdorff? (3) Is this topology finer than the standard one on R2 ? 25. Show that the product of two separated spaces is separated. 26. (1) Show that the product of two discrete spaces is discrete. (2) Show that the product of two indiscrete spaces is indiscrete. 27. (1) Show that usual R \ Q is separable. (2) Show that usual C is separable.

Let (X, d) be a metric space. ). Example. In a discrete metric space, the only convergent sequences are the eventually constant ones. Theorem. In a separated (Hausdorff ) topological space, a sequence cannot converge to two different points. 10), the previous result implies Corollary. In a metric space, if a sequence converges, then its limit is unique. We now give a fundamental and very practical result in metric spaces: Theorem. Let (X, d) be a metric space. Let A ⊂ X be non-empty. Then we have: (1) x ∈ A ⇐⇒ ∃xn ∈ A : xn −→ x.

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