I need help in order to prove the assertion in Willard Exercise 3B ...

The assertion in Willard Exercise 3B reads as follows:

I am assuming that Willard is assuming the usual topology and metric in \(\displaystyle \mathbb{R}^2\) ... and so a set...

Frontiers (Boundaries) in the Plane ...]]>

I need help in order to prove Theorem 3.11 Part 1-a using the duality relations between closure and interior ... ..

The definition of interior and Theorem 3.11 read as follows:

Readers of this post...

Closure and Interior as Dual Notions ... Proving Willard Theorem 3.11 using Duality ... ...]]>

Question : I want to find a demonstration and process this result without using the idea of circle or perpendicularity. I have been trying for a few days to find a similar demonstration of this result, possibly using...

Proof of Schiffler Point using affine geometry concepts]]>

I need help in order to fully understand an aspect of the proof of Theorem 3.7 ... ..

Theorem 3.7 and its proof read as follows:

In the above proof by Willard we read the following:

" ... ... First note that if \(\displaystyle A...\)

Closure in a Topological Space ... Willard, Theorem 3.7 ... ...]]>

I need help in order to fully understand Example 3.2(d) ... ..

Example 3.2(d) reads as follows:

and Example 2.7(e) reads as follows:

In Example 3.2(d) we read the following:

" ... It is pseudometrizable since it is the...

The Trivial or Indiscrete Topology is Pseudometrizable ... Willard, Example 3.2(d) ...]]>

Croom's definition reads as follows:

... and Singh's definition reads as follows:

The two definitions appear different ... ...

Croom requires that each

while Singh...

Local Basis in Topology ... Definitions by Croom and Singh ... ...]]>

I need help in order to fully understand M & M's proof of Lemma 3.2 (ii) ...

Lemma 3.2 (plus the definition of continuity) reads as follows:

The proof of Lemma 3.2 (ii) reads as follows:

In the above proof we read the...

Continuity in Topological Spaces ... M & M, Lemma 3.2 (ii) ... ...]]>

I need help in order to fully understand the order topology ... and specifically Example 1.4.4 ... ...

Example 1.4.4 reads as follows:

In order to fully understand Example 1.4.4 I decided to take \(\displaystyle X = \{ a, b, c \}\) where \(\displaystyle a \leq b, a \leq c\) and \(\displaystyle b \leq c\) ...

The Order Topology ... ... Singh, Example 1.4.4 ... ...]]>

I need help in order to fully understand some remarks by Singh just before he defines a sub-basis ... ..

The relevant text reads as follows:

I am unsure of Singh's arguments concerning the nature of \(\displaystyle \mathcal{ T } ( \mathcal{S} )\) ... ...

Singh writes the following:

" ... ... Clearly...

Subbasis for a Topology ... Singh, Section 1.4 ... Another Question ... ...]]>

I need help in order to fully understand some remarks by Singh just before he defines a sub-basis ... ..

The relevant text reads as follows:

To try to fully understand the above text by Singh I tried to work the following example:

\(\displaystyle X = \{ a, b, c \}\) and \(\displaystyle \mathcal{S} = \{ \{ a \}...\)

Subbasis for a Topology ... Singh, Section 1.4 ...]]>

I need some fuuther help in order to fully understand Example 4.1.1 ...

Example 4.1.1 reads as follows:

In the above example from Singh we read the following:

" ... ...Then the complement of \(\displaystyle \{ x_n \ | \ x_n \neq x \text{ and } n = 1,2, ... \}\) is a nbd of \(\displaystyle x\). Accordingly...

Convergence in Topological Spaces ... Singh, Example 4.1.1 ... ...Another Question ...]]>

I need help in order to fully understand Example 4.1.1 ...

Example 4.1.1 reads as follows:

In the above example from Singh we read the following:

" ... ...no rational number is a limit of a sequence in \(\displaystyle \mathbb{R} - \mathbb{Q}\) ... ... "

My question is as follows:

Why exactly is it the...

Convergence in Topological Spaces ... Singh, Example 4.1.1 ... ...]]>

I need help in order to fully understand Singh's proof of Theorem 1.3.10 ...

Theorem 1.3.10 (plus the definition of boundary) reads as follows:

In the above proof by Singh we read the following:

" ... ... Conversely, if \(\displaystyle x \in \overline{A} - A\), then \(\displaystyle x \in \overline{A} \cap...\)

Boundary of Subset in a Topological Space ... Singh Theorem 1.3.10 ... ...]]>

I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to date ... see below ... )

Theorem 1.3.7 reads as follows:

In the above proof by Singh we read the following:

" ...

Limit Points and Closure in a Topological Space ... Singh, Theorem 1..3.7 ... ...]]>

I need help in order to formulate a rigorous proof of

Proposition 1.3.2 reads as follows:

Can some please help me to formulate a formal and rigorous proof of Proposition 1.3.2 (a)...

Interior of a Topological Space ... Singh, Proposition 1.3.2 (a) ... ...]]>

I need some help in order to fully understand Kalajdzievski's definition of a closed set in a topological space ...

The relevant text reads as follows:

As I understand it many closed subsets of the underlying set \(\displaystyle X\) of a topological space...

Closed Subsets in a Toplogical space ...]]>

I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ...

I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...

Definitions 8.1 and 8.2 read as follows: ... ...

In the above text we read the following:

" ... ... Then one can prove that \(\displaystyle f\) is continuous iff...

Definitions of Continuity in Topological Spaces ... Sutherland, Defintions 8.1 and 8.2 ... ...]]>

It says

Show that there is (exists) a parametric curve $$\gamma :\Bbb{R}\rightarrow \Bbb{R}^2 $$ which is differentiable such that $$\gamma ( \Bbb{R})=\left\{(x,|x|),x \in \Bbb{R}\right\}$$

Can be this true? That exersice i noticed from other years notes that he is always putting it without solving it.]]>

This is should be a simple one. I know what I'm looking for but I don't know what to call it.

I have a member on PHF that I am talking with and I'd like to know the terminology used in the next two definitions.

1) I want to compactify the real numbers by defining \(\displaystyle - \infty\) and \(\displaystyle \infty\) to belong to the compactified set such that, for any member "a" in the compactified set we have that \(\displaystyle -\infty \leq a \leq...\)

Topology Terminology]]>

Let $C$ be a circle. I want to prove the following:

- If a chord of the circle changes the position keeping its length, then the midpoint is contained in a circle $C'$ concentric to $C$.
- If two chords of $C$ have their midpoints on a concentric circle $C'$, then they are equal.

I have done the following:

- Let $O$ be the centre of $C$.

Let $D$ be the midpoint of a chord passing through a point $P$.

Since $D$ is the midpoint of the chord...

Prove the statements about chords]]>

I am looking at the following question:

Which are the most important concepts that are included at the pythagorean theorem and how can we convince the students for their importance?

Are the concepts meant to be right angled triangle, hypotenuse, etc? Or what are the most important concepts?

For the importance, do we have to say the use of the pythagorean theorem?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side...

Important concepts of pythagorean theorem]]>

Show that: \(\frac{d}{dt} \int_S \bar{N} B^\alpha_\alpha dS=0\)

You will enjoy this exercise as it draws upon virtually all of the surface identities, including the Codazzi equation.

I have found the following partial solution online:

\(\begin{aligned}

\frac{d}{dt} \int_S \bar{N} B^\alpha_\alpha...

Integral of normal curvature over a closed genus 0 surface]]>

(It goes on to define the term "quotient map.")

I have a difficulty with the use of \(\displaystyle f^{-1}()\) which I think is best demonstrated by...

Quotient map question]]>

I'm taking all of this to be that \(\displaystyle E \subseteq A \subseteq X\) and that X is simply \(\displaystyle \mathbb{R} ^2\) (with it's usual topology.)

I...

Interior of a set]]>

1.) Am I correct to understand that the Einstein tensor notation used throughout the book is out of fashion and doesn't get used anymore? I don't see any of it on mathhelpboards.

Here is the Riemann Christoffel tensor in Einstein notation:

\(\nabla_i \nabla_j T^k - \nabla_j \nabla_j T^k = R^k_{mij}T^m \)...

Einstein Notation - out of fashion?]]>

I need help with an aspect of the proof of Theorem 3.3.14 ...

Bases for Tangent Soaces and Subsaces ... McInerney Theorem 3.3.14 ... ...]]>

I need help with an aspect of the proof of Theorem 3.3.13 ... ...

Theorem 3.3.13 (together with a relevant definition) reads as follows:

...

Tangent Spaces and Subspaces ... McInerney Theorem 3.3.13 ...]]>

For each positive integer $n$ let $X_n = \left\{1, 2, 3, ..., 2^n \right\}$ with the discrete topology and let $f_n : X_{n+1} → X_n$ be the function defined by:

$f_n(i) = i$ for $1 ≤ i ≤ 2^n$

$f_n(i) = 2^{n+1} − i + 1$ for $2^n < i ≤ 2^{n+1}$

Then

$X = lim_{←} \left\{X_i, f_i \right\}_{i=1}^{\infty}$

is homeomorphic to the Cantor set.

I've been trying to find a homeomorphism for this for way too long... anyone know what it would be?

I can take care of proving it's a...

Cantor Set Homeomorphic to Inverse Limit Space]]>

Here the two point set would have the discrete topology.

The product has the product topology.

And the Cantor set has the subspace topology.

I defined my function $f$ to be,

$f: C \rightarrow X$

$\sum_{n=1}^{\infty} \frac{a_n}{2^n} \rightarrow...

Cantor Set Homeomorphic to Product Space]]>

In Section 3.3 McInerney defines what is meant by a parametrized set ... and then goes on to give some examples ...

... see the scanned text below for McInerney's definitions and notation ...

I need help with...

Parametrized Set ... McInerney, Example 3.3.3]]>

Suppose that $\left\{ X_n \right\}_{n=1}^{\infty}$ is a sequence of compact, Hausdorff spaces and for each $n, f_n : X_{n+1} \rightarrow X_n$ is a continuous function (not necessarily onto).

Show that:

$X = lim_{\leftarrow} \left\{ X_n, f_n \right\}_{n=1}^{\infty} \neq \emptyset$

Furthermore, show that $X$ is compact.

I have seen a proof for a general inverse limit system, with $D$ being its directed set. However, I assume the proof differs in the problem I've stated...

Inverse Limit of Compact/Hausdorff Spaces is Nonempty and Compact]]>

Upper Semi-Continuous Collections, filling up the half-plane]]>

I need some help in order to fully understand the vector space of alternating multilinear functions ...

The relevant text from Shifrin reads as follows:

In the above text from Shifrin we read the following:

" ... ... In particular, if \(\displaystyle T \in {\bigwedge}^k ( \mathbb{R}^n )^{ \ast }\), then for any...

Vector Space of Alternating Multilinear Functions ... Shifrin, Ch. 8, Section 2.1 ... ...]]>

I am reading Chapter 13: Differential Forms ... ... and am currently focused on Section 13.1 Tensor Fields ...

I need some help in order to fully understand some statements by Browder in Section 13.1 ... ...

Section 13.1 reads as follows:

In the above text we read the following:

" ... ... We observe also that...

Differential Forms ... Another question ... Browder, Section 13.1 ...]]>

Suppose that $X$ and $Y$ are each compact sets. Consider $X \times Y$.

Let $g_x = \left\{x\right\} \times Y$ and $G = \left\{g_x : x \in X\right\}$.

Show that if $g_x \in G$ and $U \subset X \times Y$ open with $g_x \subset U$, then $\exists V$ open in $X \times Y$ such that:

(i) $g_x \subset V$

(ii) for any $g_x' \in G$, if $g_x' \cap V \neq \emptyset$ then $g_x' \subset U$

(meaning, $G$ is an upper semi-continuous collection)

If anyone has any hints to give I would...

Upper Semi-Continuous Collection]]>

The above proposition gives the wedge product of k linear functions as a determinant ...

Walschap in his book: "Multivariable Calculus and Differential Geometry" gives the definition of a determinant as follows:

From Tu's proof above we can say that ...

\(\displaystyle \text{det} [ \alpha^i (...\)

Wedge Product and Determinants ... Tu, Proposition 3.27 ... ...]]>

In the above text from Browder we read the following:

" ... ... A differential form of degree \(\displaystyle r\) (or briefly an \(\displaystyle r\)-form) in \(\displaystyle U\) is a map \(\displaystyle \omega\) of \(\displaystyle U\) into \(\displaystyle { \bigwedge}^r V ( \mathbb{R}^{ n \ast } )\) ... ... "

In other words if...

Differential Forms and Tensor Fields ... Browder, Section 13.1 ...]]>

I have a further question concerning an aspect of the proof of Theorem 1.3.1 ...

Theorem 1.3.1 and its proof read as follows:

Equation 1.3.1 in the proof implies that

\(\displaystyle \text{det} ( v_1, \cdot \cdot \cdot , v_n ) = \text{det} \left( \ \begin{bmatrix} a_{ 11} \\...\)

Permutations and Determinants ... Walschap, Theorem 1.3.1 ... ... Another question ...]]>

I need help with an aspect of the proof of Theorem 1.3.1 ...

The start of Theorem 1.3.1 and its proof read as follows:

I tried to understand how/why

\(\displaystyle \text{det} ( v_1, \cdot \cdot \cdot , v_n ) = \sum_{ \sigma } a_{ \sigma (1) 1 } , \cdot \cdot \cdot , a_{ \sigma (n) n } \ \text{det} ( e_{ \sigma (1)...\)

Permutations and Determinants ... Walschap, Theorem 1.3.1 ... ...]]>

I need help in order to fully understand Tu's Proposition 3.21 ... ...

Proposition 3.21 reads as follows:

In the above proof by Tu we read the following:

" ... ...

... \(\displaystyle = \sum_{ \sigma_{ k + l } } ( \text{ sgn } \sigma ) f( v_{ \sigma \tau (l+1) }, \cdot \cdot \cdot , v_{ \sigma \tau (l+k) }) g ( v_{ \sigma \tau...\)

Anticommutativity of the Wedge Product ... ... Tu, Proposition 3.21 ... ...]]>

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need some help in order to fully understand the proof of Proposition 12.2 on pages 277 - 278 ... ...

Proposition 12.2 and its proof read as follows:

In the above proof by Browder (near the end of the proof) we read the following:

" ... ... To see also that...

Tensors and the Alternation Operator ... Browder, Proposition 12.25 ... ...]]>

I need help in order to fully understand Tu's section on the wedge product (Section 3.7 ... ) ... ...

The start of Section 3.7 reads as follows:

In the above text from Tu we read the following:

" ... ... for every permutation \(\displaystyle \sigma \in S_{ k + l }\), there are \(\displaystyle k!\) permutations \(\displaystyle \tau\) in \(\displaystyle S_k\) that permute the...

The Wedge Product ... Tu, Section 3.7 ... ...]]>

---

A curve on a surface is geodesic iff it's geodesic curvature is zero everywhere, so I understand that I've to show that $\displaystyle k_g = \ddot \gamma \cdot (n \times \dot \gamma) = 0 $ (where n is the unit normal to $\sigma$), which I've trouble...

A curve is a geodesic on a surface proof]]>

I'd like to make sure my proof for (i) is okay, and need some clarification on the proof for (ii) that was given.

Suppose that $X'$ is an uncountable, well-ordered set and put the order topology on $X'$. Then either

Case 1: There is a point $p$ so that $\{ x:x<p \}$ is uncountable.

Case 2: If $p \in X'$ then $\{ x:x<p \}$ is...

Minimal Uncountable Well-Ordered Set]]>

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need some help in order to fully understand the proof of Proposition 12.22 on page 276 ... ...

The relevant text reads as follows:

In the above proof by Browder we read the following:

" ... ... the Kronecker delta, and hence, using Proposition 12.20...

Space of Alternating Tensors of Rank r ... ... Browder, Proposition 12.22 ...]]>

I am currently reading Chapter 12: Multilinear Algebra ... ...

I have another question regarding the proof of Theorem 12.20 on page 275 ... ...

The relevant text reads as follows:

At the end of the above proof by Browder we read the following:

" ... ... Finally, (d) is proved in exactly the same way as (a). ... ... "

Despite the above statement by...

The Alternation Operator ... Another Question ... Browder, Proposition 12.20 ... ...]]>

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need some further help in order to fully understand the proof of Theorem 12.20 on page 275 ... ...

The relevant text reads as follows:

In the above proof by Browder we read the following:

" ... ... \(\displaystyle ^{\tau }{ ( A \alpha ) } = ^{\tau }{ ( } \sum_{ \sigma \text{ in } S_r } \varepsilon (...\)

The Alternation Operator ... Browder, Proposition 12.20 ... ...]]>

Suppose that $X = lim_{\leftarrow}\left\{X_i, f_i\right\}_{i=1}^{\infty}$ is an inverse limit space so that there is an integer $N$ and for each $n ≥ N$ the function $f_n$ is an onto homeomorphism. Then $X$ is homeomorphic to $X_N$.

It seems simple that the map we are looking for is

$f: X \rightarrow X_N$

for $P=\left\{P_i\right\}_{i=1}^{\infty} \in X, f(P)=P_N$

(However I may be wrong and it wasn't so simple...)

I have defined the...

Inverse Limit Space Homeomorphism]]>

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need some further help in order to fully understand the proof of Theorem 12.9 on pages 271-272 ... ...

The relevant text (Theorem 12.8 together with the preceding definition, Definition 12.8) reads as follows:

In the above proof by Browder we read the...

A Basis for the space of covariant tensors ... ...Another Question ... Browder Theorem 12.9 ...]]>

Hello, we can use that $K=\dfrac{eg-f^2}{EG-F^2}$. Yes?]]>

I am currently reading Chapter 12: Multilinear Algebra ... ...

I am uncertain regarding a formal and rigorous proof that the tensor product is associative ... so I will give my attempt at a proof and then I hope that someone will kindly critique the proof for me ...

I will use the definitions and notation of Sections 12.7 and 12.8 ...

Sections 12.7 and 12.8 read as follows...

Proof that the tensor product is associative .... Browder Section 12.7 and 12.8 ... ...]]>

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need help in order to fully understand the proof of Theorem 12.9 on pages 271-272 ... ...

The relevant text (Theorem 12.8 together with the preceding definition, Definition 12.8) reads as follows:

In the above text from Browder, at the start of the...

A Basis for T^r, the space of covariant tensors of rank r on V ... ... Browder Theorem 12.9 ...]]>

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need help in order to fully understand the definition and nature of the set of covariant tensors of rank \(\displaystyle r\) ... as described in Browder, Section 12.7 and 12.8 ...

Te relevant text reads as follows:

My questions related to the above text are as...

Set of Covariant Tensors ... Browder Sections 12.7 and 12.8 ...]]>

I need help in order to fully understand Tu's section on the multilinear functions and k-tensors... ...

Section 3.3 reads as follows:

At the beginning of Section 3.3: Multilinear Functions, Tu writes the following:

" ... ... Denote by \(\displaystyle V^k = V \times \ ... \ \times V\) the Cartesian product of \(\displaystyle k\)...

Multilinear Functions and Alternating k-tensors ... ... Tu, Section 3.3]]>

I am working through Tu's "An Introduction to Manifolds" (I saw someone else here is looking at this book too! ) and I just want to make sure I am

(An Introduction to Manifolds) Prove that the Map is an Isomorphism of Vector Spaces]]>

No point of $X$ belongs to two elements of $G$.

What we know:

$G$ is upper semi-continuous $\implies$ if $g \in G$ and $U$ is an open set containing $g$, then there is an open set $V$ containing $g$ such that each member of $G$ which intersects $V$ lies in $U$

$G$ "fills up" $X \implies X=\cup G$

$X$ is Hausdorff $\implies \forall x_1, x_2 \in X, \exists U_1, U_2$ open...

Upper Semi-Continuous Collections]]>

I need help with Question 2.4 (a) (i) concerned with computing a directional derivative ...

Question 2.4, including the preceding definition of a directional derivative, reads as follows:

My question/problem is as follows:

In question 2.4 (a) (i) we are...

Computing the Directional Defivative ... Fortney, Question 2.4 (a) (i) ... ...]]>

I've been reading through my General Relativity text and the solution for the Schwarzschild problem. I noted a missing piece of how the metric was derived.

Can the metric have discontinuities? I'm reasonably certain it can't be discontinuous, though the Schwarzschild solution shows there can be what I would call a "removable singularity" ie. it can be removed by a choice of coordinates. But could the metric have discontinuities in its...

Metric discontinuity]]>

This is a problem I thought of myself, and I’d like to be sure that my solution is all right. Let $X,Y,Z$ be topological spaces, with continuous functions $f_1,f_2:X\to Y$ and $g_1,g_2:Y\to Z$. I wish to show that the compositions $g_1\circ f_1:X\to Z$ and $g_2\circ f_2:X\to Z$ are homotopic.

Now $f_1,f_2$ homotopic $\implies$ there is a homotopy $H_f:X\times I\to Y$ (where $I$ is the closed interval $[0,\,1]$) with $H_f(x,0)=f_1(x)$ and...

Homotopy of composition of homotopic functions]]>

When we take derivatives we use covariant derivatives to keep everything nice and tensorial. But how do you take a "covariant integral?" All I know how to do is the...

GR and integration]]>

What properties does the Manifold space have that Euclidean space doesn't have?]]>

(2) Is there an example to manifold that is not a function?

(3) Is there function that is not manifold?]]>

I need help in order to fully understand Definition 3.3 ... ...

Definition 3.3 reads as follows:

In the above text from Tapp we read the following:

\(\displaystyle df_p(v) =...\)

Directional Derivative ... Tapp, Definition 3.3 ...]]>

I need help in order to fully understand exactly how/why equations (3.4) and (3.5) follow ... ...

The relevant portion of Tapp's text reads as...

Directional Derivative ... Tapp,Equations (3.4) and (3.5) ... ...]]>

I need help in order to fully understand the proof of Proposition 1.25 ... ...

Proposition 1.25 and its proof read as follows:

My questions are as follows:

In the above text from...

Reparametrization of Curves ... Tapp, Section 4, Ch. 1 ... ...]]>

I am currently focused on Chapter 3: Submanifolds of Euclidean Spaces ... ... and in particular Section 3.2: Manifolds with Boundary ...

I am having trouble understanding Shastri's Remark following Definition 3.2.1 ...

Definition 3.2.1 and the remark following read as follows:

I am...

Manifolds with Boundary - Shastri Definition 3.2.1 and Remark 3.2.1 ...]]>

Currently I am focused on Chapter 3: Differential Manifolds ...

I need help in order to fully understand Example 3.1.13 ...

Example 3.1.13 reads as follows:

My questions are as follows:

In the above Example from Lovett we read the following:

" ... ... The image of \(\displaystyle \vec{X}\) is a half-torus \(\displaystyle M\) with \(\displaystyle y \ge...\)

Manifolds with Boundary - Lovett, Example 3.1.13 ... ...]]>

Definition of Perfectly Compact:

The point set M is said to be perfectly compact if and only if it is true that if G is a monotonic collection of non-empty subsets of M then there is a point p that is a point or a limit point of every every element of G.

Showing Compactness]]>

We are allowed to assume that X is Hausdorff and at most (if we need it) metric.

This is my first time seeing "perfectly compact"... do you approach this using...

Perfectly Compact and countable basis implies compactness]]>

I am working on constructing a $T_3$ space that is not $T_4$.

I've started with $X$, the upper half plane.

$L=\left\{(p,q):q=0\right\}$, the x-axis.

And the basis for the topology are open balls centered at points $(p,q), q>0$ intersected with $\left\{(x,y):y>0\right\}$,

and tangent discs centered at $(p,\epsilon)$ of radius $\epsilon$ union with $(p,0)$.

My professor gave some "steps" to follow. I've shown that

The Tangent Disc Space, Application of Baire Category Theorem]]>

If X is a locally compact Hausdorff space, then X is not the union of countably many nowhere dense sets.

I've tried working on this problem a couple times and I always seem to get nowhere or go in a circle.

In class we have not yet mentioned or learned anything about Baire spaces. So I am not sure if I am supposed to use the property of a Baire space (countable union of open dense sets their intersection is dense).

Our professor encouraged us to use the following...

Version of Baire Category Theorem]]>

If X is a regular Hausdorff space and X is first countable at the point P. Then there is a local basis $\left\{B_i\right\}^{\infty}_{i=1}$ at P so that for each $n \in \Bbb{N}$ we have:

$\overline{B_{n+1}} \subset B_n$.

X first countable at P $\implies \exists$ a countable local basis $\left\{A_i\right\}^{\infty}_{i=1}$ at P.

Let $B_n(P) = A_1 \cap A_2...

First Countability, Continuous f:[a,b]->[0,1]]]>

Let X be the upper half plane: $X = \left\{(x, y) \in E^2 : y \geq 0 \right\}$.

Define a basis for the topology as follows. If $P = (p, q)$ and $q > 0$ then if $\epsilon > 0$ then the set

$\left\{(x, y) \in E^2 : \sqrt{(x - p)^2 + (y - q)^2} < \epsilon\right\}\cap\left\{(x, y) : y > 0\right\}$

is a basis element; if $P = (p, q)$ and $q = 0$ then the set

$\left\{(x, y) \in E^2 : \sqrt{(x - p)^2 + (y - q)^2} < \epsilon\right\}\cap\left\{(x, y) : y > 0\right\}\cup\left\{(x...

Upper Half Disc Topology, Hausdorff not Regular]]>

Let $(X,\leq,T_<)$ be a well ordered set with the order topology.

Take H, K closed in X, $H\cap K = \emptyset $

First, assume that neither of these sets contains the least or greatest element of X.

Then,

$\forall h \in H \subseteq (X-K)...

The Order Topology on a Well Ordered Set is Normal]]>

$\newcommand{\reg}{\text{reg}}$

$\newcommand{\sing}{\text{sing}}$

$\newcommand{\set}[1]{\{#1\}}$

$\newcommand{\R}{\mathbf R}$

A

If a polynomial vanishes on a substantial portion of an irreducible variety then it vanishes...]]>

Show that if $N=2m+1$, then $U_\epsilon(C)$ is dense in $\mathcal{C}(X,\mathbb{R}^N)$.

I am given the following hint:

Given $f\in \mathcal{C}(X,\mathbb{R}^N)$ and $\delta,\epsilon >0$ choose $g:C\to \mathbb{R}^N$ so that: $d(f(x),g(x))<\delta$ for $x\in C$, and $\Delta(g)<\epsilon$. Extend...

Exercise from Munkres' Topology.]]>

$\displaystyle X:=\{0,\ 1\}^\omega=\prod\limits_{n\in\mathbb{Z_+}}\{0,\ 1\}$

So $X$ consists of the infinite sequences $\displaystyle(x_n)_{n\in\mathbb{Z_+}}$, where for each $k\in\mathbb{Z}_+$, the $k$th term $x_k$ is either $0$ or $1$. Equip $X$ with the product topology.

Show that $X$ is compact (you may not use Tychonoff’s theorem).]]>

I want to prove the incommensurability at an equilateral triangle.

The way of such a proof is the following:

1) We are looking for a "common measure" of two lines $ a $, $ b $, i.e. a line $ e $ that measures both $ a $ and $ b $ integer (i.e. there are natural numbers $ m $, $ n $with $ a = me, b = ne $).

2) We take the shorter of the two parts (e.g., $ b $) away from the longer (e.g., $ a $) until the remaining piece $ r_1 $ is shorter than $ b $. We're taking away $ r_1 $...

Incommensurability at an equilateral triangle]]>

I get how you do this in the affine case - if I have some ellipse, say:

$$\frac{x}{a^2} + \frac{y}{b^2} = c^2$$

then I can pick a point $(0, c/b)$ and run some line $y = tx+ c/b$ through it and work out algebraically the point of intersection to get expressions for $x$ and $y$ in terms of $t$ (or, I guess, alternatively, make a change of co-ordinates $y \mapsto y + c/b$ and...

Projective Parametrization]]>

Assume that there exists m≥1, so:

${S}^{m}=S∘S∘··∘S $,where the length is m

is a contraction.

1) Show that S has a unique fixed point

2) Show that for $m=2$ we can say that $S=cos:[0,\frac{π}{2}]→[\frac{π}{2}]$

Let X = (X,d) be a metric space.

A map $S: X → X$ is contraction if there exists a number $0≤β≤1$ so:

$d(S_x,S_y )≤βd(x,y)$ for all $x,y∈X$

A fixed...

Complete metric space and unique fixed point]]>

Let $\gamma: (\alpha,\beta)\to\mathbb{R}^3$ be a regular curve with torsion and curvature that are never $0.$

Show that $\gamma$ is a generalized helix curve if and only if the binormal $b(s)$ makes a fixed angle with a constant vector $a \in \mathbb{R}^3, a\not=0.$

Generalized helix curve]]>

1. Let X and Y be topological spaces; let $p: X \to Y$ be a surjective map. The map p is said to be a

2. If a set X is a (topological) space and if $p: X \to Y$ is a surjective map, then there exists exactly one topology T on Y relative to which p is a quotient map; it is called the

Quotient Maps]]>

He (and I've seen this description elsewhere also) says that when the bug sits on one of the horizontal edges (technically it's sitting on both) and looks up or down, it sees 'a little half disk'. Similarly, the bug sitting on one of the corners (all the corners) sees 'four little quarter disks'...

Square torus, again]]>

A vector field $X$ is said to be conformal if $L_Xj=0$ where j is the almost complex structure. The conformality condition is equivalent to $j\beta =\beta j$. Where $\beta : ker(\theta) \mapsto ker(\theta)$ such that $\beta(u)= \nabla_uX$ and $\theta$ is contact form such that $\theta(X)=1$. \\

my question is how can i see that $j\beta =\beta j$ is equivalent to $\beta^*+\beta$ being a multiple...

Equivalence of the following two terms]]>

Given a polyhedron (doesn't have to be regular), let $F_n$ be the number of $n$-gon faces, and let $V_n$ be the number of vertexes at which exactly $n$ edges meet. Verify the following (there are several similar relations, but I'll just give one):

$(2V_3 + 2V_4 + 2V_5 + ...) - (F_3 + 2F_4 + 3F_5 + ...) = 4$.

I don't understand what we mean when we

Euler's formula?]]>

Mosers worm problem is the problem of finding a shape of smallest area that can cover all curves of unit length. The shape is allowed to be rotated and translated to cover the curve. (Make a blanket of minimal size that can cover a worm of length 1dm). I work with a version of the problem where the shape is required to be convex.

I wonder if anyone knows a result of the following type:

There is a smallest shape W where W contains the points (0,0) and (0,1) and W is contained in the...

Mosers worm problem. I need a theorem.]]>

A subspace of a Banach space is complete if and only if it is closed.]]>

Let $ABC$ be a triangle and let $A'\in BC$, $B'\in AC$ and $C'\in AB$. I want to show the following:

If $A',B',C'\neq A,B,C$ and either zero or two of the points $A',B',C'$ are at one side of triangle, then $A',B',C'$ are collinear iff $\frac{|AC'|}{|C'B|}=\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=1$.

Could you give me a hint how we could show that? ]]>

$A=X\cap F_1$ where $F_1$ is closed. Furthermore, $B=Y\cap F_2$ where $F_2$ is closed. It follows that $X$ and $Y$ are closed. Thus,

$

\begin{eqnarray*}

A\times B &=& (X\cap F_1)\times (Y\cap F_2)\\

&=& (X\times Y)\cap (F_1\times F_2)\\

&=& \text{closed...

Closed set relative]]>

$\tau_Y=\left\{U\subseteq Y:\bigcup U =\left(\bigcup_{ {[a]\in U} }[a]\right)\in\tau_X\right\}$

be written

$\tau_Y=\left\{U\subseteq Y:\bigcup U =\left(\bigcup_{ {[a]\in U} }\left\{[a]\right\}\right)\in\tau_X\right\}$

instead?

since $\bigcup U=\bigcup_{ [a]\in U }\left\{[a]\right\}$

or is this not correct?

Wolfram mathworld writes it as $\bigcup_{ [a]\in U } a$ which doesn't make any sense to me. I've never really felt comfortable with these formulas.]]>

Wikipedia https://en.wikipedia.org/wiki/Connected_space#Path_connectedness here claims a finite topological space is connected if and only if it is path-connected. Perhaps this is because connectedness and local path-connectedness imply path-connectedness and finite topological spaces are locally path-connected.

an...

when is a finite topological space path-connected?]]>

$\overline{A}=\{x\in X: U\cap A\ne \varnothing, \text{ for all } U\in\tau \text{ and } x\in U\}.$

Hello, if $x\in\overline{A}$, then for all $V$ neighborhood of $x$, we have $V\cap A\ne \varnothing.$ My question is if its neccesary the inclusion of sets for showed.]]>

(Writing $I$ to denote the closed unit interval, the

I tired the following. Think of $S^1$ as...

The Join of Two Circles Is the Three Sphere]]>

Now if I take another case where I change $B_1=\mathbf{R}$-$\{3\}\in \tau_1$ and keep...

Comparison of Topologies:basic]]>