Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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an infinite discrete subspace

**Each infinite subspace of a KC space contain an infinite discrete subspace.** Proof: Let $ (X,\tau)$ be aKC space, and $A ‎‎\subseteq‎ X$ is infinite. since $A$ does not have the cofine topolog , ...
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35 views

Show that the following Statements is true?

Let $\tau $ be the topology on $\mathbb R$ for which the interval $[a,b)$ form a base.Let $\sigma$ be a topplogy on $\mathbb R $ such that $\tau \subseteq \sigma$. Then If the map $ x \mapsto -x$ ...
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26 views

Confused about boundary points

Please pretend that these lines are zoomed-in parts of a disc in $\mathbb{R}^2$. The first picture on the left is a point immediately adjacent, or tangent to, the disc. The second picture on the ...
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52 views

Grothendieck topology

Hello : Here is a small parapgraph that i try to understand : If $ M $ is a smooth manifold, then we can recover the underlying set of $ M $ by considering the set $ \mathrm{Hom} ( \{ \star \} , ...
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23 views

Closure of a linear subspace of $C([a,b])$

Given the space $C([a,b])$ (the collection of all real-valued, continuous (with respect to the metric $d(x,y)=|x−y|)$ functions defined on the interval $[a,b]⊆R$), along with the uniform norm and the ...
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111 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
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35 views

Removing a line boundary form a half disk

Let's cut a disk in two halves. Take one of the two halves. Its boundary is made of two pieces: one is the half circle and the other is "the line boundary" along the cut. I was told that the half ...
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53 views

$E$ is closed $\leftrightarrow E^{c}$ is open

I'm having difficulty following this proof provided in Principles of Math. Analysis by Rudin. Pf First suppose $E^{c}$ is closed. Choose $x \in E$. Then $x \notin E^{c}$, and $x$ is not a limit ...
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28 views

Some basic Topological proof help please.

Basically im really bad at proofs and i havent done math in almost a year and decided id like to learn topology on my own... just want someone to be really critical on my solutions please also i would ...
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21 views

Stone-Čech compactification not by ultrafilters only.

I am familiar with Stone-Čech compactification using ultrafilters. But, I, somehow can't understand the construction by commutative diagram, and certainly can not see the connection between the two ...
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32 views

Fibres, rammification points and continuity

Trying to understand the relation between the following: given a function $f:\mathbb{R} \to \mathbb{R}$ with all fibres either empty or of size 3, what can be said about the continuity of such a ...
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31 views

Covering dimension of a compact metric space

I would like to see the proof of the following fact (references appreciated). A compact metric space $X$ has covering dimension $\leqslant n$ if and only if there is a continuous surjection $\pi ...
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45 views

Complex analysis winding number

We have that $f:\mathbb S^1 \rightarrow \mathbb S^1 $ and $f(z)=f(1)\widehat{\phi}(z)$ with $\widehat{\phi}(\exp{2\pi it}) = \exp(2 \pi i \phi(t)),$ where $\phi:I \rightarrow \mathbb{R} $ is a ...
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35 views

Show that $(1,1,1,…)$ is a limit point of a set $A$.

Let $X_j$ be $\{0,1\}$, the 2 point set, with the discrete topology for $j = 1,2,…$. Let $X$ be the countable product of the $X_j$'s with the product topology. Let A be the set which consists of ...
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28 views

Order topology on non discrete set

Set $X=\{1,2\} \times\mathbb Z^+$ where $\mathbb Z^+$ is positive intergers. Consider $X$ under dictionary order. The order topology on $X$ is not discrete. Why it is not discrete here? Why I cannot ...
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19 views

Separating convex sets in a tvs $X$.

I got doubt with the proof of this theorem. Let $X$ be a tvs, $A,B \subset X$ with $A$ an open convex set and $B$ convex such that $A \cap B = \emptyset$. Then there exists $f \in X^*$ (where ...
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23 views

Are all subbasis subsets of basis?

In topology, each element of basis $\{B_k\}$ can be expressed as finite intersections of elements of subbasis, i.e. $B_k=S_{n_1}\cap ...\cap S_{n_m}$ Does the meaning of "finite intersections" also ...
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28 views

Preimage of boundary

If $f$ is a continuous function between topological spaces, is it true that: $$f^{-1}(\partial A)=\partial f^{-1}(A)$$ for a subset $A$ of the domain? If not, what further requisites would I require?
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38 views

Product topology of compact subsets

Suppose K,L are compact subsets of topological spaces X,Y respectively, and that KxL < W where W is open in XxY. Prove that for each x in K there exist sets U_x, V_x open in X, Y respectively and ...
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39 views

If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous?

If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous? I am trying to prove continuity by limits but am failing: suppose ...
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34 views

Is the closure of the cells in a CW complex compact?

As the cells in a CW complex afaik are homeomorphic to the open ball by definition, I was wondering whether this also means that their closure is a compact set? And if this is true, I would be ...
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31 views

CW complex in $\mathbb{R}^n$?

I am supposed to show that a finite CW complex is homeomorphic to a compact subset of some $\mathbb{R}^n$, where $n$ is large enough. My idea was the following: A CW complex is also T3, which means ...
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25 views

Locally compact CW complex

I want to prove the folowing: A CW complex is locally compact iff every point has a neighbourhood that intersects with just finitely many cells. I already did the implication "locally compact", ...
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28 views

Why is this series uniformly convergent on compact subsets

Given the series (family of functions) for every |z|<1 ($z\in \mathbb C$) we have, $$\sum_{n=0}^\infty z^n = 1 + z + z^2 + ... = \frac1{1-z}$$ This is given as an example of a 'normal family' of ...
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30 views

A peculiar fact about 3-dimensional complex projective space

I'm working on a result for my master's thesis, that right now involves translating a proof I don't quite follow, to something that is a bit more in line with what I already know. We define ...
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20 views

Compactification

I just read a dubious line in a text about compactifications: In many cases we can think of a compactification $r(X) \subset \overline{r(X)}$ of a topological space $X$ in such a way that ...
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56 views

Video lectures about algebraic topology

I am looking for a video lectures about algebraic topology in graduate level course. If someone know some site, please let me know
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38 views

Classification of closed surfaces

I am doing a course in topology and is currently working on the classification theorem for closed surfaces. After realizing that every closed surface is either homeomorphic to the sphere or the sphere ...
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34 views

Continuity of a function in the product topoogy

Hi everyone I would like to understand if my reasoning is correct. Let $X$ be the space of sequences with values in the interval $[0,1]$, i.e. if $\mathbb{N}$ is the set of natural numbers, $x\in X$ ...
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35 views

Lebesgue number proof

This is a proof I found from Marsden's Elementary classical analysis but there's a part I don't understand. Lemma: Assume that $A$ is sequentially compact. Let ${U_i}$ be an open cover of$A$. Then ...
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21 views

Circular jump progrssion

in a circular pond, lotus petals are arranged along the perimeter. a frog leaps from one petal to another in such a way that starting from a petal, it skips one petal and jumps to the next one; then ...
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79 views

Is it false that the complement of an open set is closed?

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a continuous function. Let $Z(f)$ be the zero of $f$. Prove that $Z(f)$ is closed. This is one of problems in my mid-term exam. I have used ...
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46 views

$C(X,Y)$ metric space and compactness

I have a question regarding the metric set $C(X, Y)= \{ f : X \rightarrow Y \, | \, f \, \text{is continous and bounded}\}$. Now, suppose that $X$ is totally bounded and $Y$ is compact. If $F \subset ...
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22 views

A question about the topology of real Banach Spaces.

Let B be a real Bananach Space whose dimension is at least 2 and let S be a subset of B that is an an open ball. Is the complement of S (with respect to B) always connected?
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32 views

The discontinuous or the characteristic function is the boundary

Let $A \subseteq \mathbb{R}^n$. Put $D = \{ x \in R^n : \chi_A(x) \; \; \text{is discontinuous } \} $. Then do we have that $$ \partial A = D $$ ??? My attempt: If $x \in \partial A$, then we can ...
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35 views

How to prove that a sub-space of the functions $f: X \to Y$ is equicontinuous?

Let $X$ and $Y$ be two metric and compact spaces, and $C(X,Y)$ - the metric space of the continuous functions $f:X\rightarrow Y$. Denote by $Y^X$ the space of all functions (not just continuous) ...
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23 views

On the inverse images of morphisms between compact Riemann surfaces

The following text comes from the book Introduction to compact Riemann surfaces and dessins d'enfants. "Let $f : X \to Y$ be a non-constant morphism between compact Riemann surfaces. Let $\Sigma = ...
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18 views

What is the nth barycentric simplicial subdivision?

I read it in a paper, but, unfortunately, am not familiar with it. Here's my guess: For example, in $\mathbb{R}^2$, a simplex is a triangle, and if we divide it by three lines connecting each vertex ...
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32 views

Which one are the set of neiborhood $\mathcal{V}(x)$ for each $x\in X$?

Let $X$ a finite set, and $X^{*}=X\cup \{\omega\}$ wiht $\omega\notin X$. Given a filter $\mathcal{F}$ on $X$, Show that Which one are the set of neiborhood $\mathcal{V}(x)$ for each $x\in X$? I ...
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17 views

Determine a scalar field with gradient perpendicular to gradient of another scalar field

Given a scalar field, is it possible to find a scalar field whose gradients are perpendicular to the gradients of the original scalar field? The given scalar field is smooth.
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22 views

Find a circle which is a strong deformation retract of $\mathbb{R^2}-x_0$

Let $x_o\in\mathbb{R^2}$. Find a circle which is a strong deformation retract of $\mathbb{R^2}-x_0$ Proof: If $x_0=(a,b)$, Let $S=\{(x-a)^2+(y-b)^2=1\}$. Then $f_t(r,\theta) = (\exp((1-t)r),\theta)$, ...
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78 views

Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ ...
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17 views

Showing that the function $C(b)$ is a compact set for $|b| < 1$

I am reading "An Invitation to Dynamical Systems", and one of the challenge problems is to prove that $C(b)$ is a compact set where $C(b)$ is defined as the set of all numbers that can be expressed in ...
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18 views

Uniform convergence of $f_n(z)=\sum_{j=o}^n z^j$ on the open unit complex disk.

I have the sequence $f_n(z)=\sum_{j=o}^n z^j$ on the open unit complex disk ($\Delta$). My question is whether or not my approach is correct to the following problems: is the sequence normal? Does ...
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37 views

How can I envision the open ball around the french railways metric?

I have that the French railways metric defined by a metric space $(\mathbb{R}^2,d)$ has a distance function that is a metric as follows: $$d(x,y) = \begin{cases} \|x-y\|, & \text{if $x,y,0$ are ...
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27 views

Is the closure of the intersection of a set with a closed subspace is equal to the closure of the set intersection the subspace?

If $M$ is a closed subspace of a topological vector space $X$ and $A$ intersects $M$. Is $\overline{A\cap M}=\overline{A}\cap M$
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39 views

Showing that an identity map for a metric space in $\mathbb{R}^2$ is continuous but that its inverse isn't.

Suppose that $A=(\mathbb{R}^2,d)$ is a metric space with $d(x,y)=||x-y||$. I would like to show that if I have an identity map from $I:A \to \mathbb{R}^2$ with its euclidean distance function, then ...
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19 views

Neighborhoods for continuous functions between CG spaces

I have a couple of problems regarding the existence of certain neighborhoods, so as to prove continuity of suitable functions. Suppose then that $Y,X$ and $Z$ are compactly-generated Hausdorff spaces ...
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46 views

point set topology: closed points dense

Let $X$ be an irreducible finite dimensional Jacobson scheme (i.e. the closed points lie dense and the underlying topological space is sober). If one chooses for every closed point $x \in |X|$ an open ...
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32 views

Separability of a certain space of continuous functions

Let $I$ be a separable, locally compact Hausdorff space, and let $V$ be a separable, locally convex, complete topological vector space. Consider the function space $C(I, V)$ with the compact-open ...