![arXiv:math/9911224v2 [math.GT] 9 Dec 1999](http://s1.studyres.com/store/data/016370695_1-31d422643fefeca7faefc96120ceb1d7-300x300.png)
Introduction to higher homotopy groups and
... This can also be seen using some differential topology. Choose a point x1 ∈ S n with x0 6= x1 . Let f : (S k , p) → (S n , x0 ). By a homotopy we can arrange that f is smooth. Moreover if k < n then we can arrange that x1 ∈ / f (S k ). Then f maps to S n \ {x1 } ' Rn , which is contractible, so f is ...
... This can also be seen using some differential topology. Choose a point x1 ∈ S n with x0 6= x1 . Let f : (S k , p) → (S n , x0 ). By a homotopy we can arrange that f is smooth. Moreover if k < n then we can arrange that x1 ∈ / f (S k ). Then f maps to S n \ {x1 } ' Rn , which is contractible, so f is ...
closed sets, and an introduction to continuous functions
... be more flexible with the nature of our intervals: Lemma 8. f is continuous at x if and only if for every open interval set I containing f (x), there’s an open interval J containing x such that f (J) ⊆ I. Proof. Suppose f is continuous. Let I be an open interval with f (x) ∈ I. Then there’s some > ...
... be more flexible with the nature of our intervals: Lemma 8. f is continuous at x if and only if for every open interval set I containing f (x), there’s an open interval J containing x such that f (J) ⊆ I. Proof. Suppose f is continuous. Let I be an open interval with f (x) ∈ I. Then there’s some > ...
Another property of the Sorgenfrey line
... PROOF. Since X x Y is regular, it is enough to show that if if/" is an open cover of X x Y which is closed under finite unions, then 7i’ has a a-locally finite refinement. Let W be such a cover. Let F(n)> be a spectral 1-network for Y, as in (2.4). For each n ~ 1 and for each ntuple (03B11, ··· ...
... PROOF. Since X x Y is regular, it is enough to show that if if/" is an open cover of X x Y which is closed under finite unions, then 7i’ has a a-locally finite refinement. Let W be such a cover. Let F(n)> be a spectral 1-network for Y, as in (2.4). For each n ~ 1 and for each ntuple (03B11, ··· ...
