Intereting Posts

How prove this matrix $\det (A)=\left(\frac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}\neq 0$
If $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$, then find $f(2)$
Generalizing Bellard's “exotic” formula for $\pi$ to $m=11$
Product rule for logarithms works on any non-zero value?
How do we prove $\cos(\pi/5) – \cos(2\pi/5) = 0.5$ ?.
Fitting a closed curve on the roots of ${x \choose k}-c$
Why do both sine and cosine exist?
Upper Triangular Form of a Matrix
Group actions and associated bundles
Proof on a particular property of cyclic groups
Find $f(2a-x)$ from given equation
Why is a straight line the shortest distance between two points?
An arrow is monic in the category of G-Sets if and only if its monic the category of sets
Easy visualizations of small countable ordinals
Why are punctured neighborhoods in the definition of the limit of a function?

This is an example from Munkres’s Topology (Example 4 in Section 22 titled “The Quotient Topology”, 2nd edition).

Example 4:Let $X$ be the closed unit ball $$\{ x \times y \mid x^2 + y^2 \le 1\}$$ in $\mathbb{R}^2$, and let $X^{\ast}$ be the partition of $X$ consisting of all the one-point sets $\{ x \times y \}$ for which $x^2 + y^2 < 1$, along with the set $S^1= \{ x \times y \mid x^2 + y^2 = 1 \}$. Typical saturated open sets in $X$ are pictured by the shaded regions in the figure below. One can show that $X^{\ast}$ is homeomorphic with the subspace of $\mathbb{R}^3$ called the unit 2-sphere, defined by $$S^2 = \{ x \times y \times z \mid x^2 + y^2 + z^2 =1 \}.$$

I am confused about two points in the example.

- CW construction of Lens spaces Hatcher
- Why do we need surjectivity in this theorem?
- A bijective map that is not a homeomorphism
- Topologies of test functions and distributions
- Isotopy and homeomorphism
- Connectedness of the boundary

Problem 1:About the saturated open sets in $X$. In the example, the typical ones are pictured by the shaded regions ($U, V$) in the figure. However, I am not sure whether theboundaries(visually, by picture) of $U$ and $V$ are included. Particularly, there are two boundaries for $U$. Are they both contained in $U$? And why?

Problem 2:How to show that $X^{\ast}$ is homeomorphic with $S^2$? What is the mapping between them?

The following is my understanding:

Solution to problem 1:For $V$, the boundary is not included. For $U$, the outer boundary is included, while the inner one not. Is this solution right?

- Topology such that function is continuous if and only if the restriction is.
- About the interior of the union of two sets
- Quotient Space $\mathbb{R} / \mathbb{Q}$
- Closed Subgroups of $\mathbb{R}$
- Subsets of the reals when the Continuum Hypothesis is assumed false
- Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected.
- Formal proof that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.
- Is the result true when the valuation is trivial and $\dim(X)=n$?
- Cauchy sequence is convergent iff it has a convergent subsequence
- Does the Euler characteristic of a manifold depend upon the field of coefficients?

You’re right, Munkres has given a very confusing picture. (Also, the term *saturated* is hardly standard in this context — I’ve never seen in used in other topology books.)

For (1): you are correct about $V$. For $U$, your answer is a possible correct answer, but it would also be OK not to include the outer boundary; in that case, the corresponding open set on the sphere would look like a punctured open disk, i.e., missing a point inside.

For (2): the idea is that we can take the upper hemisphere and stretch it over the entire sphere, collapsing the equator to a single point (the south pole). To describe this formally, let the disk on the left have radius $\pi$. Use polar coordinates, so a point in the disk is $(r,\theta)$. On the sphere, use coordinates $(\alpha,\beta)$ where $\alpha$ is the angle of latitude, but measured from the north pole (so $\alpha=0$ at the north pole, $\alpha=\pi$ (i.e., $180^\circ$) at the south pole), and $\beta$ is the angle of longitude.

To be quite explicit about the definition of $\alpha$: In the diagram latitude, the latitude is the angle ϕ. However, you’ll note that ϕ is measured up from the equator. That’s standard, of course. I want an angle like ϕ, but measured down from the north pole. So in the northern hemisphere, α=π/2−ϕ (using radians, π/2=90∘), and in the southern hemisphere, α=ϕ+π/2.

Now let $(r,\theta)$ map to the point with $\alpha=r, \beta=\theta$. When $r=\pi$, we have the entire circumference mapping to the south pole.

- How to construct this Laurent series?
- How can you pick the odd marble by 3 steps in this case?
- For which topological spaces $X$ can one write $X \approx Y \times Y$? Is $Y$ unique?
- Are positive real numbers $x,y$ allowed to be taken out during this proof?
- Show $\Omega$ is simply connected if every harmonic function has a conjugate
- Ring isomorphism for $k(G \oplus \mathbb Z )$ with $G$ torsion-free and abelian
- Is the Collatz conjecture in $\Sigma_1 / \Pi_1$?
- Are there $3$ disjoint copies of $2K_{3,3} \cup (K_{5,5} \setminus C_{10})$ in $K_{11,11}$?
- What is the simplest way to prove that the logarithm of any prime is irrational?
- Why is every positive integer the sum of 3 triangular numbers?
- What map of ring spectra corresponds to a product in cohomology, especially the $\cup$-product.
- Subtraction of a negative number
- How to compute this finite sum $\sum_{k=1}^n \frac{k}{2^k} + \frac{n}{2^n}$?
- application of strong vs weak law of large numbers
- $R^n$ can be decomposed into a union of countable disjoint closed balls and a null set