Topology Exercise sheet 3
... 1. Suppose that X = R with the usual metric topology. We denote by Q the set of all rational numbers. Find the set of all limit points for the following sets A: (a) A = Q (b) A = R \ Q (c) A = (−1, 1) ∪ (1, 1) (d) A = {xn : n ∈ N} where xn is any convergent sequence. (e) (Ill-defined and harder) Wha ...
... 1. Suppose that X = R with the usual metric topology. We denote by Q the set of all rational numbers. Find the set of all limit points for the following sets A: (a) A = Q (b) A = R \ Q (c) A = (−1, 1) ∪ (1, 1) (d) A = {xn : n ∈ N} where xn is any convergent sequence. (e) (Ill-defined and harder) Wha ...
Topology Exercise sheet 5
... There are many questions: you do not need to do them all but you should think whether you could do them. Ideally, if I asked you in the tutorial how to do a question you would be able to answer it. They are in no particular order so if you can’t do one go on to the next. 1. Suppose that A ⊂ R is not ...
... There are many questions: you do not need to do them all but you should think whether you could do them. Ideally, if I asked you in the tutorial how to do a question you would be able to answer it. They are in no particular order so if you can’t do one go on to the next. 1. Suppose that A ⊂ R is not ...
1. Connectedness of a metric space A metric (topological) space X is
... Let X be a metric (topological) space and A be a subset of X. Then A is also a metric (topological) space of A. Every open set in A is of the form U ∩ A for some open set U of X. We say that A is a (dis)connected subset of X if A is a (dis)connected metric (topological) space. Theorem 1.2. Let X and ...
... Let X be a metric (topological) space and A be a subset of X. Then A is also a metric (topological) space of A. Every open set in A is of the form U ∩ A for some open set U of X. We say that A is a (dis)connected subset of X if A is a (dis)connected metric (topological) space. Theorem 1.2. Let X and ...
Finish Metric Spaces. Interlude on Quotient
... Example Let X = [0, 1] (with std. top.) and define x ∼ y if x = y or if x = 0 and y = 1 or if x = 1 and y = 0. (So, the only distinct points of X which are equivalent are 0 and 1). Let T denote the unit circle, as a subspace of R2 . I claim that we can define a homeomorphism from X/ ∼ to T. First we ...
... Example Let X = [0, 1] (with std. top.) and define x ∼ y if x = y or if x = 0 and y = 1 or if x = 1 and y = 0. (So, the only distinct points of X which are equivalent are 0 and 1). Let T denote the unit circle, as a subspace of R2 . I claim that we can define a homeomorphism from X/ ∼ to T. First we ...
Topology, MM8002/SF2721, Spring 2017. Exercise set 3 Exercise 1
... • Show that if X and Y are Hausdorff, so is X × Y . • Show that if X and Y are first countable, so is X × Y . • Show that if X and Y are second countable, so is X × Y . Which of these statements hold for all finite products? Which for arbitrary products? Exercise 7. Consider an arbitrary product of ...
... • Show that if X and Y are Hausdorff, so is X × Y . • Show that if X and Y are first countable, so is X × Y . • Show that if X and Y are second countable, so is X × Y . Which of these statements hold for all finite products? Which for arbitrary products? Exercise 7. Consider an arbitrary product of ...
Algebraic approach to p-local structure of a finite group: Definition 1
... Algebraic approach to p-local structure of a finite group: Definition 1 (Puig?) Let G be a finite group and S be a Sylow subgroup of G. The fusion system of G is the category FS (G), whose objects are the subgroups of S, obF = {P ≤ S}, and whose morphism sets are given by HomF (G)(P, Q) = HomG(P, Q) ...
... Algebraic approach to p-local structure of a finite group: Definition 1 (Puig?) Let G be a finite group and S be a Sylow subgroup of G. The fusion system of G is the category FS (G), whose objects are the subgroups of S, obF = {P ≤ S}, and whose morphism sets are given by HomF (G)(P, Q) = HomG(P, Q) ...
NOTES ON FORMAL SCHEMES, SHEAVES ON R
... Exercise 5. Check that this is equivalent to the definition in [Lur10, lecture 11]. Alternatively, show that they are different and correct these notes. Let G be a formal group over R. The abelian group structure induces a unit map e : SpecR R → G. The Lie algebra g of G is e∗ TG — a line bundle on ...
... Exercise 5. Check that this is equivalent to the definition in [Lur10, lecture 11]. Alternatively, show that they are different and correct these notes. Let G be a formal group over R. The abelian group structure induces a unit map e : SpecR R → G. The Lie algebra g of G is e∗ TG — a line bundle on ...
