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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Topology of $L^2$ space

Cardinality of space of all funcions $f: \mathbb R \rightarrow \mathbb R$ is $\beth_2$. However, cardinality of space of all such square-integrable functions, space $L^2$, is $\beth_1=\mathfrak c$, ...
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1answer
25 views

Proving $g(y)f^{''}(x)+f(x)g^{''}(y)+ (\alpha ^2 + \beta ^2)f(x)g(y)= 0 \implies f^{''}(x)+\alpha ^2f(x) = 0 , g^{''}(y)+ \beta ^2 g(y)=0$

I was reading a proof in physics, suddenly I'm stuck at proving this passage: $g(y)f^{''}(x)+f(x)g^{''}(y)+ (\alpha ^2 + \beta ^2)f(x)g(y)= 0 \implies f^{''}(x)+\alpha ^2f(x) = 0 , g^{''}(y)+ \beta ...
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1answer
38 views

Questions about sigma-algebra

I am learning measure theory this semester. The definition for sigma-algebra is "a collection of sets that is closed under complements and countable unions and intersections." I wonder what does it ...
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28 views

Two geometrical objects in same dimensional plane are homeomorphic.

What can be a good way to prove that two geometrical objects in same dimensional plane are homeomorphic?? For example....to show that a circle and a ellipse is homeomorphic in $\Bbb R^2$ and a ...
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50 views

Show that a set is connected and contractible?

I am having trouble with a practice qual exam question: Let $X = \{x,y\}$ with $\emptyset,X,\{x\}$ open. Show that X is connected and contractible? For the first part, I would assume not. That ...
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3answers
48 views

Product of Two Metrizable Spaces

I am having trouble with a practice exam question: $$\text{Show that if $X$ and $Y$ are metrizable, then so is $X\times Y$}$$ What I have so far: Given metric spaces $(X,d_x)$ and $(Y,d_y)$, I know ...
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65 views

The quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open.

I'd like to show that the quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open, where I'm considering $\mathbb P^n$ as the quotient space of $\mathbb R^{n+1} \setminus \{0\}$ ...
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31 views

Modes of convergence for continuous functions

I just wondered about what modes of convergence for continuous functions $f_n:X\rightarrow Y$ between topological spaces there are. Of course there is pointwise convergence, which is defineable for ...
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25 views

To prove existence of an open set of functions

I am trying to prove the following: In $C(X,Y)$ with $X=[0,1]$ and $Y$ of finite dimension $K$, $C(X,Y)$ having the topology of uniform convergence, for any $K$ finite there exists an open set of ...
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21 views

simplicial approximation and infinite complexes

It is well known that if $X$ is a finite simplicial complex then for every continuous map $f:|X|\to |Y|$ there exists a simplicial map $F: X^{(n)}\to Y$ that $|F|$ is homotopic to $f$. Does anyone ...
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68 views

General topology problem

Let $X$ be an infinite set. Let $T$ be a topology on $X$ such that all infinite subsets of $X$ belong to $T$. Prove that $T$ is the discrete topology on $X$. i know that all the infinite subsets of X ...
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1answer
24 views

James' theorem—going from the separable case to the general case

Consider the following famous theorem by Robert C. James (1964): Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and ...
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1answer
19 views

First countable space and $\tau \subset \tau^*$

Let $\tau$ and $\tau^*$ be topologies on X with $\tau$ coarser than $\tau^*$ i.e $ \tau \subset \tau^*$. (i) Show that if (X,$\tau^*$) first countable, then (X,$\tau$) is also first countable I have ...
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+50

Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ ...
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1answer
19 views

$X$ contains at least two points & at least one isolated point. Prove $X$ is not connected.

Can we take two sets $G_1 = (x_1)$, where $x_1$ is the isolated point, and $G_2 = B(x_2;\epsilon)-(x_2)$ where $x_2$ is a limit point and show that the set- connectedness conditions hold? Help would ...
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1answer
22 views

Locally Compact Spaces: Separation Property

Given a locally compact Hausdorff space. Every compact set has a compact neighborhood base: $$C\subseteq U:\quad N\subseteq U\quad(C\subseteq N^°)$$ My construction was contrary to Rudin's: ...
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31 views

To determine whether range of f is closed , connected etc

Let $E= \{ (x,y) : |x| + |y| \leq 1 \}$ . Define $f : E \to \mathbb R$ by $f(x, y) = x + y / 1 + x^{2} + y^{2} $ Then range of $f$ is A . Connected open set B . Connected closed set C. ...
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4answers
74 views

Why does a topology contains its basis?

The definition of a basis is given as: Suppose a collection $\mathscr B$ of subsets of a given set $X$ satisfies: $\forall x \in X, \exists B \in \mathscr{B}$ such that $x \in B$. $B_1 , B_2 \in ...
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1answer
70 views

Homeomorphisms of an open set onto itself.

Given two distinct points $a,b\in U$ where $U$ is an open subset of $\mathbb R^n$, can one always find a homeomorphism $f:U\to U$ with $f(a)=b$ or does one need any further conditions on $ U $ or the ...
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70 views

How many of these are topologies?

Let $X$ be a set with $3$ elements. The set of subsets of the power set of $X$ is $2^{2^3}$ elements. How many of these are topologies? Is there a trick to this problem, or is it just a "plug ...
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1answer
66 views

Convergence of a sequence with assumption that exponential subsequences converge?

Problem One of my best friends asked me to think about the following problem: Suppose a sequence $\{a_n\}_{n=1}^\infty$ satisfies $\lim_{n\to\infty}a_{\lfloor\alpha^n\rfloor}=0$ for each ...
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1answer
99 views

Are all functions $f:\mathbf{Z}\to\mathbf{Z}$ “continuous”?

I read the following definitions in Glen E. Bredon's "Topology and Geometry": Let $\mathbf{x},\mathbf{y}\in\mathbf{R}^n$ and $$ ...
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281 views

An Illustrated Classification of Knots.

Let me be honest here: I know very little about Knot Theory. I'm sorry. I've a friend though, someone with no training in Mathematics at all but who is a huge fan of knots (for whatever reason), who ...
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1answer
79 views

Metrics on $\mathbb R^n$, Counting continuous functions and Open sets

Given the set $\mathbb{R}^n$ with metric $d$. We define continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ by open sets -we say that function is continuous iff the pre-image of every open set ...
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1answer
50 views

Complement of a point of a Compact Connected Hausdorff Space has no compact maximal connected subspace

This question is a slight modified version of Compact Connected Hausdorff Space has no compact component in the complement of a point Let $X$ be a Hausdorff Compact Connected Space. Prove that ...
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2answers
74 views

Urysohn's Lemma: Proof

Given a normal space $\Omega$. Then closed sets can be separated continuously: $$h\in\mathcal{C}(\Omega,\mathbb{R}):\quad h(A)\equiv0,\,h(B)\equiv1\quad(A,B\in\mathcal{T}^\complement)$$ ...
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0answers
42 views

$G_{\delta}$ set

I have a homework as follows: Let $\mathcal{U}$ be an open cover of $X$.If $\mathcal{V}$ is finite subset of $\mathcal{U}$ but $\mathcal{V}$ doesn't cover $X$, prove that $X-\bigcup\mathcal{V}$ is a ...
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28 views

Weakly closed $\iff$ closed using the Separation theorem

My question is about the following problem. $X$: Banach space, $C$: convex subsets of $X$. Then, followings are equivalent. i) $C$ is closed. ii) $C$ is weakly closed. I ...
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37 views

$K$ compact set $\subset A$ , open set of $\mathbb{C}$, implies that there exists $D$(smooth boundary) s.t. $K \subset D\subset \bar{D} \subset A$

I consider this situation: I have an open set $A\subset \mathbb{C}$ and a compact set $K\subset A$ and I want to prove that there exists an open set $D$ such that $K \subset D\subset \bar{D} \subset ...
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2answers
56 views

Prove that the set $X+Y$ is closed

Consider two sets: $$X=\bigl\{(x,y)\in \mathbb R^{2}:xy=1\bigr\}$$ $$Y=\bigl\{(x,y)\in \mathbb R^{2}:|x|\le 1,|y|\le 1\bigr\}.$$ We find that , both the sets are closed. But, is the set $X+Y$ ...
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1answer
64 views

There does not exist a holomorphic map between torus and Riemann sphere

So the question is as follows. Prove that there is no meromorphic function $f$ such that at every $z\in \mathbb{C}$ we have $f(z)=f(z+1)$ and $f(z)=f(z+i)$ with only simple poles at the points $m+ni, ...
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2answers
14 views

A question about the definition of a subbasis in a topological space

In Munkres Topology he defined a subbasis for a topology on $X$ as a collection of subsets of $X$ whose union equals $X$. Does this imply that a subbasis could contain a closed subset of $X$ as an ...
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1answer
51 views

Product of a First Countable Space by a Fréchet Space

It's easy to see that the product of two First Countable spaces is First Countable, and it's easy to show that every First Countable space is a Fréchet Space (i.e, if $A \subset X$ and $p \in \bar A$ ...
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28 views

Why $\mathcal{D}(\Omega)$ be a topological vector space is important?

Let $\Omega\subset\mathbb{R}^N$ be an open set and $\mathcal{D}(\Omega)$ the set of all infinitely differentiable functions with compact support on $\Omega$. In the study of PDEs, we use the ...
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1answer
56 views

Compact Connected Hausdorff Space has no compact component in the complement of a point

Let $X$ be a Hausdorff Compact Connected Space. Prove that $X\setminus\{x\}$ can't be expressed by the disjoint union of two connected sets with one them being compact.(lets assume the empty set is ...
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2answers
33 views

Uniqueness of a limit from a general topological space to a Hausdorff space

I read a proof showing the uniqueness of the limit from a general topological space to a Hausdorff space but don't understand it. Given a function $f : S \rightarrow T$ from a topological space to a ...
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1answer
67 views

Volume of a paracompact manifold

It is stated, without proof, in Wald (1984) (General Relativity) that given any connected manifold $M$ (which is by definition paracompact), one may define a volume measure $\mu$ such that $\mu[M]$ is ...
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1answer
67 views

Group Actions: Orbit Space

Given a group action $G\curvearrowright X$. Consider the orbit space: $\pi:X\to X/G$ Do continuous group actions correspond to open projections, i.e.: $$l_g\in\mathcal{C}(X)\quad(g\in ...
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16 views

isometrically isomorphism [duplicate]

How can embed separable Banach to Cb(X)(family of all bounded continuous functions on topological space X) with non metrizable X ? If X is locally compact or Tychonof is very well. note : We know ...
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1answer
30 views

embedding projective plane in 4-space? [closed]

Is it possible to embed projective plane in 4-space? If not what is the reason and what is the smallest singularity set?
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1answer
48 views

Open neighborhoods in the definition of a manifold

At the beginning of Spivak's "A Comprehensive Introduction to Diff. Geom." (p.3), in the definition of a (topological) manifold $M$, every point $x$ has a neighborhood $U$ that is homeomorphic to ...
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1answer
25 views

Question regarding characters and point open topology

I was wondering why the following claim is correct: Let G* be the group of all continuous homomorphisms from the topological group G and the unit circle (call it T). Then G* is an intersection of a ...
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1answer
52 views

What is the topology of this quotient of $S^2 \times S^1$?

So suppose you take an $S^2$, then you put an $S^1$ fiber over it which degenerates by smoothly shrinking to a point at its poles. What is the topology of this space in more familiar terms (assuming ...
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2answers
42 views

confusion over Finite intersection property

It is stated that $A_n={(\frac{-1}{n},\frac{1}{n})}$, then arbitrary intersection of open sets need not be open is true as in this case $\bigcap_{i=1}^{\infty}=\left \{0 \right \}$ is not open. Now ...
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1answer
24 views

A topological space with a transitive action.

Let $X$ be a topological space on which a topological group $G$ acts transitively. Given $x\in X$ let $$stab(x)=\{g\in G\;|\; gx=x\}.$$ I want to show that $X$ is homeomorphic to $G/stab(x)$ for any ...
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1answer
39 views

Is this a topological closure operation?

Does any relation $\propto\,\subseteq X\times \mathcal P(X)$ that extends $'\!\!\in'$ in the way that: $x\in M\Rightarrow x\propto M$ $\neg\exists x\in X:x\propto\emptyset$ $x\propto A \subseteq B ...
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The Seifert-Van Kampen theorem as a push-out

My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following. Let $X$ be a topological space, and $U, V\subseteq X$ two ...
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What is the domain and image of the composition of mappers in a manifold

I was trying to understand the following: which I got from: http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf I was wondering, why the domain was: $$ \alpha(U_{\alpha} \cap ...
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27 views

Contractibility of complex manifold

I'm trying to show that for $v^2 = w^4 - a^4$ for real $a$ and complex $v, w$ that this manifold deform retracts to a point $(0, a)$ but can't seem to figure out a path that remains on the surface. ...
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1answer
50 views

If M is a manifold of dimension $ n \neq0$ then M has no isolated points.

I am in doubt whether the following statement is true or false: "If M is a manifold of dimension $ n \neq0$ then M has no isolated points." The idea that made me find the true statement was as ...