By Hajime Sato

The only so much tricky factor one faces whilst one starts to benefit a brand new department of arithmetic is to get a think for the mathematical feel of the topic. the aim of this ebook is to assist the aspiring reader collect this crucial good judgment approximately algebraic topology in a quick time period. To this finish, Sato leads the reader via basic yet significant examples in concrete phrases. in addition, effects aren't mentioned of their maximum attainable generality, yet by way of the easiest and so much crucial situations.

In reaction to feedback from readers of the unique version of this ebook, Sato has further an appendix of beneficial definitions and effects on units, basic topology, teams and such. He has additionally supplied references.

Topics coated contain primary notions akin to homeomorphisms, homotopy equivalence, basic teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute periods. gadgets and examples thought of within the textual content comprise the torus, the Möbius strip, the Klein bottle, closed surfaces, phone complexes and vector bundles.

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**Extra info for Algebraic Topology: An Intuitive Approach**

**Example text**

5 Example. Let (X, τ ) be an indiscrete space and A a subset of X with at least two elements. Then it is readily seen that every point of X is a limit point of A. ) The next proposition provides a useful way of testing whether a set is closed or not. 6 Proposition. Let A be a subset of a topological space (X, τ ). Then A is closed in (X, τ ) if and only if A contains all of its limit points. Proof. We are required to prove that A is closed in (X, τ ) if and only if A contains all of its limit points; that is, we have to show that (i) if A is a closed set, then it contains all of its limit points, and (ii) if A contains all of its limit points, then it is a closed set.

Y .......... .......... 4 are satisfied. Thus B is indeed a basis for the euclidean topology on R2 . 9 we defined a basis for the euclidean topology to be the collection of all “open rectangles” (with sides parallel to the axes). 5 shows that “open rectangles” can be replaced by “open equilateral triangles” (with base parallel to the X-axis) without changing the topology.

X ∈ X is said to be a limit point (or accumulation point or cluster point) of A if every open set, U , containing x contains a point of A different from x. 2 Example. Consider the topological space (X, τ ) where the set X = {a, b, c, d, e}, the topology τ = {X, Ø, {a}, {c, d}, {a, c, d}, {b, c, d, e}}, and A = {a, b, c}. Then b, d, and e are limit points of A but a and c are not limit points of A. Proof. The point a is a limit point of A if and only if every open set containing a contains another point of the set A.