Compactness and compactification
... Compactness is a powerful property of spaces, and is used in many ways in many different areas of mathematics. One is via appeal to local-to-global principles; one establishes local control on some function or other quantity, and then uses compactness to boost the local control to global control. An ...
... Compactness is a powerful property of spaces, and is used in many ways in many different areas of mathematics. One is via appeal to local-to-global principles; one establishes local control on some function or other quantity, and then uses compactness to boost the local control to global control. An ...
The Free Topological Group on a Simply Connected Space
... that for any k-group G and any continuous pointed map f of X into G, there exists a unique continuous homomorphism ft of F(X) into G such that fJi = f. It is proved in Ordman [11] and Hardy [7] that F(X) always exists, is independent of the choice of base point, and is algebraically the free group o ...
... that for any k-group G and any continuous pointed map f of X into G, there exists a unique continuous homomorphism ft of F(X) into G such that fJi = f. It is proved in Ordman [11] and Hardy [7] that F(X) always exists, is independent of the choice of base point, and is algebraically the free group o ...
Domain of sin(x) , cos(x) is R. Domain of tan(x) is R \ {(k + 2)π : k ∈ Z
... has its values in − π2 , π2 has its values in [0 , π] ...
... has its values in − π2 , π2 has its values in [0 , π] ...
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 ...
Key
... We must first determine a consistent means of finding the angle between two intersecting planes. Angles are formed by two lines. We need to find a representative line from each plane. An arbitrary line will not work. For example we could pick the line of intersection for one of the planes. Then the sec ...
... We must first determine a consistent means of finding the angle between two intersecting planes. Angles are formed by two lines. We need to find a representative line from each plane. An arbitrary line will not work. For example we could pick the line of intersection for one of the planes. Then the sec ...
Homework Solution Section 2.3 8. Applying Theorem 2.4, we check
... R2 − 3R1 ; R3 ← R3 − R1 ; R3 ← R3 − 4R2 . So, the homogeneous system has infinitely many solutions given by α = −3γ, β = 4γ, γ ∈ <. Hence the three vectors are linearly One particular ...
... R2 − 3R1 ; R3 ← R3 − R1 ; R3 ← R3 − 4R2 . So, the homogeneous system has infinitely many solutions given by α = −3γ, β = 4γ, γ ∈ <. Hence the three vectors are linearly One particular ...