By Haynes R. Miller, Douglas C. Ravenel

Through the iciness and spring of 1985 a Workshop in Algebraic Topology used to be held on the college of Washington. The direction notes via Emmanuel Dror Farjoun and by means of Frederick R. Cohen contained during this quantity are rigorously written graduate point expositions of sure facets of equivariant homotopy thought and classical homotopy concept, respectively. M.E. Mahowald has incorporated a few of the fabric from his extra papers, signify quite a lot of modern homotopy idea: the Kervaire invariant, strong splitting theorems, computing device calculation of risky homotopy teams, and stories of L(n), Im J, and the symmetric teams.

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How does this topology compare with the usual topology for the reals ? This topology is called the lower limit topology. There is a similarly defined upper limit topology. 4. Topologize the real numbers by chosing as a base the family of all semi-infinite intervals of the form (a, r ° ° ) = {x : x > a] for any real numbers. Verify that this is a base for a topology and describe the open sets. What is the closure of a set in this topology ? How does this topology compare with the usual, upper limit, and lower limit topologies for the reals ?

T h i s topology, of course, is t h e usual topology. A n o t h e r base for t h e usual topology would be t h e collection of all open intervals with rational e n d p o i n t s . I t will be i m p o r t a n t later t h a t this is a base with only a countable n u m b e r of elements. As an example of a subbase we could choose all sets which are either of t h e form {χ : χ > a} or t h e form {χ : χ < a} for some real n u m b e r a. Again it is sufficient to let a be an arbitrary rational n u m b e r .

4. Topologize the real numbers by chosing as a base the family of all semi-infinite intervals of the form (a, r ° ° ) = {x : x > a] for any real numbers. Verify that this is a base for a topology and describe the open sets. What is the closure of a set in this topology ? How does this topology compare with the usual, upper limit, and lower limit topologies for the reals ? We will call this topology the right-hand topology. Theie is a similarly defined left-hand topology for the reals. 5. If y is a subbase for a topology Τ, show that Τ is the intersection of all topologies containing the family if.