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Symmetry, Broken Symmetry, and Topology in Modern Physics: A First Course

Mike Guidry M., Y. (Yang) Sun

Chapter 24

Topology, Manifolds, and Metrics - all with Video Answers

Educators

Chapter Questions

02:40

Problem 1

Consider a set $M=\{a, b, c, d, e\}$ and the collection of subsets
$$
\tau \equiv\{\theta, M,\{a\},\{c, d\},\{a, c, d\},\{b, c, d, e\}\},
$$

Tony Ni
Tony Ni
Numerade Educator

Problem 2

Consider a set $M=\{a, b, c, d\}$ and a collection of subsets $\tau=\{\emptyset, M,\{a\},\{b \mid\}$. Show that $\tau$ is not a topology on $M$.

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Problem 3

Prove that an interval without endpoints is homeomorphic to the real number line $\mathbb{R}$. Thus, boundedness is not a topological invariant. Hint: Take $X=\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and $Y=\mathbb{R}$, and consider the map $f: X \rightarrow Y$ given by $f(x)=\tan x . * * *$

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Problem 4

What is the first homotopy group of a two-dimensional torus $T^2 ?$ Hint: For the 2-torus, $T^2=S^1 \times S^1, *=$

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02:08

Problem 5

Using an open covering corresponding to the family of concentric open disks
$$
S_\alpha=\left\{(x, y): x^2+y^2<\left(1-\frac{1}{1+\alpha}\right)\right\},
$$
where $\alpha=1,2, \ldots$, show that the open unit disk of Fig. $24.5(\mathrm{~b})$ is not compact. *t *

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
01:59

Problem 6

Show that the winding number $Q$ of Eq. (24.7) gives $Q=n$ for the mapping (24.6).

Suzanne W.
Suzanne W.
Numerade Educator

Problem 7

From Problems 2.9 and 2.11 , the group $\mathrm{D}_2=\{e, a, b, c\}$ has a factor (quotient) group with respect to the abelian invariant subgroup $H=\{e, a\}$,
$$
\mathrm{D}_2 / H=H+M=\{e, a\}+\{b, c\}
$$
with a map $\phi$ from $D_2$ to $D_2 / H$ given by
$\varphi:$
Show that if the space for $D_2$ is equipped with a topology defined by the open sets
$$
\tau=\{\emptyset,\{e\},\{a \mid,\{b \mid,\{c\},\{e, a, b, c\}\},
$$
the inverse $\operatorname{map} \phi^{-1}$ implies that the quotient space has a topology also. ***

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Problem 8

Are the following homeomorphic? (a) A closed interval and an open interval for the real numbers $\mathbb{R}$ ? (b) A parabola and a hyperbola? (c) A circle $S^1$ and $\mathbb{R}$ ?

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01:05

Problem 9

Sketch the results of path multiplications (indicated by $x$ ) for these examples:
where each loop is defined in the same $2 \mathrm{D}$ euclidean plane with a single hole.

Jennifer Stoner
Jennifer Stoner
Numerade Educator