By O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, and V. M. Kharlamov

This textbook on simple topology features a distinct advent to common topology and an advent to algebraic topology through its such a lot classical and hassle-free section founded on the notions of basic crew and protecting house. The booklet is customized for the reader who's decided to paintings actively. The proofs of theorems are separated from their formulations and are accrued on the finish of every bankruptcy. This makes the ebook seem like a natural challenge e-book and encourages the reader to imagine via each one formula. A reader who prefers a extra conventional variety can both locate the proofs on the finish of the bankruptcy or bypass them altogether. This type additionally caters to the professional who wishes a instruction manual and prefers formulations no longer overshadowed via proofs. lots of the proofs are easy and simple to find. The e-book could be worthy and stress-free for readers with particularly assorted backgrounds and pursuits. The textual content is dependent in one of these means that you possibly can verify what to anticipate from every bit and the way to take advantage of it. there's center fabric, which makes up a comparatively small a part of the booklet. The middle fabric is interspersed with examples, illustrative and coaching difficulties, and correct discussions. The reader who has mastered the center fabric acquires a robust heritage in easy topology and should suppose at domestic within the setting of summary arithmetic. With nearly no must haves (except actual numbers), the publication can function a textual content for a direction on common and starting algebraic topology.

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It is also called trivial topology . ) It consists only of X and 0 . B. This i s a topological structure, i s it not? Here are slightly less trivial examples. 2 . 1 . Let X be the ray [0, + oo), and let n consist of with a 2 0. Prove that Q is a topological structure . 2 . 2. Let X be a plane. Let L: consist of origin. Is L: a topological structure? 0, 0, X , and all rays (a, + oo) X , and all open disks centered at the 2 . 3. Let X consist of four elements: X = {a, b,c, d} . e. , satisfy the axioms of topological structure : 4Thus, n is important: it is called by the same word as the whole branch of mathematics.

It is also called trivial topology . ) It consists only of X and 0 . B. This i s a topological structure, i s it not? Here are slightly less trivial examples. 2 . 1 . Let X be the ray [0, + oo), and let n consist of with a 2 0. Prove that Q is a topological structure . 2 . 2. Let X be a plane. Let L: consist of origin. Is L: a topological structure? 0, 0, X , and all rays (a, + oo) X , and all open disks centered at the 2 . 3. Let X consist of four elements: X = {a, b,c, d} . e. , satisfy the axioms of topological structure : 4Thus, n is important: it is called by the same word as the whole branch of mathematics.

Show that the property of being closed is not local. 5. G Transitivity of Induced Topology. Let (X, n) be a topological space, X :J A :=J B . Then (nA ) B = ns , i. e. , the topology induced on B by the relative topology of A coincides with the topology induced on B directly from X. 5. 7. Let (X, p) be a metric space, A c X . Then the topology on A generated by the induced metric P IA x A coincides with the relative topology induced o n A by the metric topology on X . 5. 8. Riddle. The statement 5 .