# A second course in general topology by Heikki Junnila By Heikki Junnila

Similar geometry and topology books

Geometry 2

This can be the second one a part of the 2-volume textbook Geometry which supplies a truly readable and full of life presentation of huge components of geometry within the classical feel. an enticing attribute of the e-book is that it appeals systematically to the reader's instinct and imaginative and prescient, and illustrates the mathematical textual content with many figures.

Schaum's Outline of Descriptive Geometry

This publication presents a radical figuring out of the elemental stages of graphical research for college kids of engineering and technology. It additionally prepares scholars to unravel tougher difficulties of this kind encountered later of their person fields. lively studying is inspired and examine time lowered with a number of difficulties solved step by step.

Modern Classical Homotopy Theory

The middle of classical homotopy idea is a physique of principles and theorems that emerged within the Fifties and was once later mostly codified within the proposal of a version classification. This middle comprises the notions of fibration and cofibration; CW complexes; lengthy fiber and cofiber sequences; loop areas and suspensions; etc.

Extra resources for A second course in general topology

Sample text

Then we have that N = X. Let x ∈ X N . Then Nx ∈ N ˆ. N It follows from Propositions 9-11 that if a T1 -space X has a normal closed base, then X has a Hausdorff compactification and hence X is Tihonov. We shall next show that the converse holds: every Tihonov space has a normal closed base. The zero-set of a function f : X → R is the set f −1 {0}. A zero-set of a space X is the zero-set of a continuous function X → R. We denote by ZX the collection of all zero-sets of X. If f : X → R is continuous, then so is the function g, defined by the rule g(x) = min(1, |f (x)|).

Then d can be extended to a continuous pseudometric of X provided that there exists a continuous pseudometric ρ of X such that we have d ≤ ρ on A2 . Proof. Assume that ρ is as above. Define r : X 2 → R by setting r(x, y) = d(x, y), if (x, y) ∈ A2 ρ(x, y), if (x, y) ∈ A2 , and note that we have r ≤ ρ on X 2 . For all x, y ∈ X, denote by δ(x, y) the infimum of the numbers n i=0 r(xi , xi+1 ), where n ∈ N, xi ∈ X for each i, x0 = x and xn+1 = y. It is easy to see that δ is a pseudometric of X. Moreover, since r ≤ ρ, we have that δ ≤ ρ and hence δ is continuous.

Let d be a pseudometric of X, and let A ⊂ X. We denote by d(A) the d-diameter sup{d(x, y) : x, y ∈ A} of A. For x ∈ X and ∅ = A ⊂ X, we denote by d(x, A) the d-distance inf{d(x, a) : a ∈ A} of the point x to the set A. If X is a space and d is a continuous pseudometric of X, then for every non-empty A ⊂ X, the function x → d(x, A) is continuous. This is a consequence of the inequality |d(x, A) − d(y, A)| ≤ d(x, y) which is valid for all x, y ∈ X. We mention some ways of obtaining “new pseudometrics from old”.