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Preliminaries

Preliminaries. http://cis.k.hosei.ac.jp/~yukita/. Topological concepts. Topological space X each point x is enclosed in a family of subsets of X , called its neighborhoods . A subset A of X is arbitrarily near a point x of X

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Preliminaries

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  1. Preliminaries http://cis.k.hosei.ac.jp/~yukita/

  2. Topological concepts • Topological spaceX • each point x is enclosed in a family of subsets of X, called its neighborhoods. • A subset A of X is arbitrarily near a point x of X • every neighborhood of x contains at least one member of the subset A. • f : XY is continuous • whenever a subset A of X is arbitrarily near a point x of X, • f(A) must be arbitrarily near f(x) in Y.

  3. Neighborhoods of a point x in E2 These are not neighborhoods. Neighborhoods

  4. Arbitrarily near x in E2 Not near. These sets are arbitrarily near x.

  5. Relative topology of a set in E2

  6. Homeomorphic spaces Disc with one hole Sphere with two holes Cylinder Annulus

  7. Inequivalent spaces Sphere Torus

  8. Cut and glue Circle and Trefoil are homeomorphic. They are differently embedded in space. Circle Arc Trefoil knot

  9. Cut and glue Cylinder and Doubly-twisted strip are homeomorphic. They are differently embedded in space. Doubly-twisted strip Cylinder Moebius strip The border of the cylinder consists of two disjoint circles. The border of the Moebius strip is a single circle.

  10. Homeomorphic spaces Sausage-link space Pin-cushion space

  11. Inequivalent spaces Wire handcuff Wire eight The wire handcuff has two points that have T-shaped neighborhoods.; in the wire eight no point has a T-shaped neighborhood and one point has some X-shaped neighborhoods.

  12. Thickened versions are homeomorphic. Thickened handcuff

  13. Homeomorphic. Enlarged punctured torus Punctured torus

  14. Which pair is homeomorphic?Count the number of circles on the border. Disc with non-overlapping hands Sphere with three holes Disc with overlapping hands

  15. Homeomorphic to the solid sphere with two handles Thickened punctured torus Thickend disc with non-overlapping hands Thickend sphere with three holes

  16. S1 x I Cylinder Interval of circles Circle of intervals

  17. S1 x S1 Circle of circles Torus Circle of circles

  18. Accumulation points

  19. Does the set A accumulate at x? Ａ Ａ Ａ ｘ ｘ ｘ

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