2(a) Let R be endowed with standard topology. Show that for all x ε
... Let Y be a subspace of X. If U is open in Y and Y is open in X. Show that U is open in X. 7marks x Show that the mapping f: R → R+ defined by f(x) = e is a homeomorphism from R to R+ . ( Recall that a homeomorpism from one topological space to another is a bijective function f such that f and f -1 a ...
... Let Y be a subspace of X. If U is open in Y and Y is open in X. Show that U is open in X. 7marks x Show that the mapping f: R → R+ defined by f(x) = e is a homeomorphism from R to R+ . ( Recall that a homeomorpism from one topological space to another is a bijective function f such that f and f -1 a ...
1. Natural transformations Let C and D be categories, and F, G : C
... Example 1.1. The homomorphism ∂∗ : Hn (X, A) → Hn−1 (A) is a natural transformation. It is not a natural isomorphism. Example 1.2. In Bredon IV.3 we defined for each pointed topological space (X, x0 ) a homomorphism of abelian groups π1 (X, x0 )/[π1 , π1 ] → H1 (X). This gives a natural transformati ...
... Example 1.1. The homomorphism ∂∗ : Hn (X, A) → Hn−1 (A) is a natural transformation. It is not a natural isomorphism. Example 1.2. In Bredon IV.3 we defined for each pointed topological space (X, x0 ) a homomorphism of abelian groups π1 (X, x0 )/[π1 , π1 ] → H1 (X). This gives a natural transformati ...
Categories and functors
... whose objects are the points of X and whose arrows x - y are the homotopy classes of paths from x to y. Digression 1.10 You might have noticed that in many categories A, the sets A(A, B) carry extra structure. For instance, if A = k-Mod then they are abelian groups, and if A is a suitable category o ...
... whose objects are the points of X and whose arrows x - y are the homotopy classes of paths from x to y. Digression 1.10 You might have noticed that in many categories A, the sets A(A, B) carry extra structure. For instance, if A = k-Mod then they are abelian groups, and if A is a suitable category o ...
Classifying spaces and spectral sequences
... The space BG if often a classifying space for G in the usual sense, as one can see as follows. Consider the category G with ob(G)==G and with a unique isomorphism between each pair of elements ofG, i.e. mor(G)=GxG. It is equivalent to the trivial category with one object and one morphism, so BG is c ...
... The space BG if often a classifying space for G in the usual sense, as one can see as follows. Consider the category G with ob(G)==G and with a unique isomorphism between each pair of elements ofG, i.e. mor(G)=GxG. It is equivalent to the trivial category with one object and one morphism, so BG is c ...
Homework sheet 6
... that what you’ve written down is a morphism. 4. If X is a topological space, {Ui }i∈I is an open cover of X, and Z is a subset of X, prove that Z is closed if and only if Z ∩ Ui is closed in Ui (when Ui is given the induced topology) for all i. [This verifies a claim made in class.] 5. Let C be a sm ...
... that what you’ve written down is a morphism. 4. If X is a topological space, {Ui }i∈I is an open cover of X, and Z is a subset of X, prove that Z is closed if and only if Z ∩ Ui is closed in Ui (when Ui is given the induced topology) for all i. [This verifies a claim made in class.] 5. Let C be a sm ...
