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  1. Sep 2024
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    1. chrisaldrich 30 Sep 2024
      Z orn ’s lemma. Suppose S, < is a partially ordered set with the property that every chain in S has an upper bound. Then S contains amaximal element.

      typo : < should be ≤

      Zorn's lemma lemmas partially ordered sets axiom of choice definitions
    2. chrisaldrich 30 Sep 2024
      The axiom o f choice. Suppose {Si}, i E I, is a family of nonemptysets. Then there is a function / from I into U / Si such that f(i) E Sifor each i e I.

      For any collection of non-empty sets, one can create a set by choosing one element from each set in the given collection.

      There are a variety of other equivalent ways to state this as well as names. One variation is Zorn's lemma.

      axiom of choice Zorn's lemma definitions
    3. chrisaldrich 30 Sep 2024

      Gemignani, Michael C. Elementary Topology. 2nd ed. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley Publishing Company, 1971.

      Annotations URL<br /> Alternate annotations URL

      topology Michael C. Gemignani References Michael Miller UCLA Extension mathematics metric spaces homeomorphisms separation axioms convergence continuity countability compactness connectedness homotopy theory fundamental group Tychonoff theorem
    4. chrisaldrich 30 Sep 2024
      opology has reached the point where a mathematician engagedin topological research is not only justified in calling himself a topologist,but he must specify whether he is a point set topologist, differentialtopologist, algebraic topologist, or some other topological specialist.

      sub-branches of topology: - point-set topology<br /> - differential topology<br /> - algebraic topology

      topology point-set topology differential topology algebraic topology
    5. chrisaldrich 30 Sep 2024
      Geometrically, topologywas the study of properties preserved by a certain group of transformations, the homeomorphisms. Geometry itself can be considered as thestudy of properties preserved by certain types of functions; e.g., Euclideanmetric geometry is the study of properties preserved by rigid (that is,distance-preserving) transformations (known sometimes as congruences).(Of course, as with topology, it is somewhat unfair to try to define geometry as the study of one particular thing.)
      topology definitions congruences homeomorphisms

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