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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Sequence Lemma explanation

Then every neighbourhood $U$ of $x$ contains a point of $A$. So I don't see it happening unless $X$ is a metric space, but the proof is for any topological space.
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Continuous mapping problem

I have always confused on various "continuous mapping" problem. So here it is: Let $f:X_1 \rightarrow X_2$, $f$ is continuous. Then: if $X_1$ is open, is $X_2$ open? Similarly, if $X_1$ closed, is ...
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27 views

an interesting topology question about open sets

Suppose we are in $\mathbb{R}^n$ and say $\mathcal{B}$ is the collection of all open sets of $\mathbb{R}^n$ : all the open balls. we know $\mathcal{B}$ is a basis for $\mathbb{R}^n$. Now, put $$ T : ...
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20 views

Why do we care about non-$T^0$ spaces?

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure "recognizes" that different points are different: No two points have exactly the same ...
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12 views

Exercises Topological Spaces Schaum

Prove that ($\ R^2$, T) is a topological space where the elements of T are $\phi $ and the complements of finite sets of lines and points
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39 views

A proof of a small topological lemma

I just stumbled upon a proof of topological lemma that I don't understand: it would be great if anyone could give me some advices. To be blunt, I am convinced that the proof does work but to me it ...
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2answers
42 views

Exercise of topological spaces [duplicate]

$X$ is an infinite set and $T$ topology of $X$ in which all the infinite subset of $X$ are open, prove that $T$ is the discrete topology of $X$
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22 views

Tangent Bundles to manifolds

I am having trouble trying to visualize exactly what a tangent bundle to the klein bottle is spuposed to look like. Is it possible for one to decompose it as a direct sum of simpler bundles?
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Explain why the open mobius band is a smooth surface [on hold]

Explain why the open mobius band is a smooth surface and find a homeomorphic copy of it inside the real projective space RP^2 and inside the Klein Bottle K
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1answer
15 views

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$.

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$. $[X,Y]=\{f:X\to Y,f$ continuous $\}/\sim$ where $\sim$ is the homotopic equivalence. ...
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1answer
10 views

Disconnecting a complex vector space

Can a (complex) dimension $n$ subspace disconnect a (complex) dimension $n+1$ vector space ? If the answer is no, what if we replace "vector space" by "manifold" ?
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1answer
16 views

Topology of metric completion of Euclidean metric

Lets consider $\cal{M}=\mathbb{R}^{2}\backslash\{(0,y)\}\text { with } \{|y|\le1\}$ with the Euclidean metric with line element $ds^{2}=dx^{2}+dy^{2}$. Now consider the distance function given by ...
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Prove that $C_1$ and $C_2$ are homotopic fixing endpoints.

Let $C_1$ and $C_2$ be two great circles in $S^2$, intersecting at the points $p,q$. If we consider $C_1$ and $C_2$ as curves starting and ending at $p$. Prove that $C_1$ and $C_2$ are homotopic ...
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1answer
23 views

The family of open intervals that do not contain $0$

Let $T$ be the collection of all open sets in $\mathbb{R}$ not containing $0$ union $\mathbb{R}$ i.e $$T=\{(a,b)\subset\mathbb{\bar R}:0\notin(a,b)\}\cup\{\mathbb{R}\}$$ Then what is true about $T$? ...
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22 views

Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
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1answer
26 views

Continuity of function proof

Let $f:X \to Y \times Z$ be given by $f(x)=\bigl( f_{1}(x), f_{2}(x) \bigr)$. Prove that $f$ is continuous iff $f_{1}$ and $f_{2}$ are continuous. I'm struggling to relate the pre image of $h$ ...
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1answer
32 views

String through 1 hole 3-torus

Okay So I had stayed up way too late thinking about this problem and I typed my question wrong. The question is: How do I deform a 3 dimensional 1 hole torus to go around a line? ...
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3answers
178 views

Homeomorphism(topological spaces) version of Cantor–Bernstein–Schroeder theorem

Let $A$ , $B$ be topological spaces such that there for some subset $D$ of $B$ there is a homeomorphism form $A$ to $D$ and for some subset $E$ of $A$ there is a homeomorphism form $B$ to $E$ ; then ...
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1answer
18 views

Inverse limit of countable (or even finite) sets

Sorry if this is kind of a stupid question. I am trying to wrap my head around inverse limits. Question : can an inverse limit of countable sets be uncountable ? Typically something like a Cantor set. ...
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1answer
22 views

Showing the right half of the unit hyperbola is a complete metric space.

Let $f : \mathbb{R} \rightarrow \mathbb{R}^2$ be given as follows. $$f(\theta) = (\cosh \theta, \sinh \theta)$$ I want to argue that $\mathrm{im}(f)$ is a complete metric space with respect to the ...
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32 views

Relation about Disk and Sphere

Definition of sphere and disk are following \begin{align} S^n =\{ (x_1 , \cdots x_{n+1}) \in \mathbb{R}^{n+1} | \sum x_i^2 =1 \} \end{align} \begin{align} D^n =\{ (x_1 , \cdots x_{n}) \in ...
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2answers
37 views

Minimal $T_0$-topologies

Let $X$ be an infinite set and let $\tau$ be a $T_0$-topology on $X$. Does $\tau$ contain a $T_0$-topology that is minimal with respect to $\subseteq$?
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245 views

Does every homogeneous space allow a group structure?

Let $(X,\tau)$ be a homogeneous space, that is for all $x,y \in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there a group operation $*:X\times X\to X$ such that ...
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44 views

How to denote the inside of a manifold?

In $\mathbb{R}^3$, I want to denote the inside of a closed surface $S$. Now I could define a volume $V$ such that $A = \partial V$, but I do not want to introduce an unnecessary, additional symbol $V$ ...
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24 views

Example of $x$ being adherent point but not accumulation point?

So I was just reading Apostol and I see that if $x$ is an accumulation point of set $S$, it has to be an adherent point as well. I guess it's possible for $x$ to be an adherent point only, not an ...
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1answer
38 views

Dual notion of the subspace topology

Let $X$ and $Y$ be sets with $\iota:X\to Y$ an injection. If $Y$ is a topological space, we define the subspace topology on $X$ as the initial topology induced by this diagram. Analogously, if $X$ ...
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24 views

set of all regular values

Let $M$ be a compact manifold and $f: M\longrightarrow \mathbb{R}$ be smooth. Show that the set of all regular values of $f$ is open. How can I prove it? Could someone help me?
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Let $(X,d)$ be a metric space, $C$ a compact subset, and $K$ a closed subset. Prove that $K \cap C = \emptyset$ iff $d(K,C) > 0$.

Let $(X,d)$ be a metric space, $C$ a compact subset, and $K$ a closed subset. Prove that $K \cap C = \emptyset$ iff $d(K,C) > 0$. For this problem I was going to consider $d(x,F) = \inf d(x,y)$ ...
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Is every connected subset of the Sierpiński triangle arcwise connected?

I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming $S$ is the connected subset) $S$ ...
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Find Int and cl of set.

Let $C[0,1]$ be space of continuous functions defined on euclidean segment $[0,1]$ with values all over euclidean line $\mathbb{R}$ with "supremum" metric. $d_{sup}(f,g)=sup\{|f(x)-g(x)| : x \in ...
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38 views

Proof verification of compactness

Let $K$ be the set $\{0\} \cup \{1/n : n \text{ is an element of the positive integers}\} $ Prove that $K$ is compact. In my head, it seems that what they are asking in this question to prove is ...
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1answer
53 views

prove function is not continuous

Let $f$ from the reals to the reals be given by $f(x)=4+x$ if $x$ is rational and $f(x)=4-x$ if $x$ is irrational. Prove $f$ is not continuous where $R$ has standard topology . Let $U$ be an open ...
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2answers
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The name for the quotient property.

We call a surjective $f:X\rightarrow Y$ a quotient mapping if it satisfies, for every $U\subset Y$ (continuity, continuous) $U$ is open $\Rightarrow$ $f^{-1}(U)$ is open and (???) ...
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7 views

Gluing of two geodesic space along a proper space is geodesic.

Let $X_1$ and $X_2$ geodesic metric space glued along A a proper subspace of both and then given the pseudo metric. Why is the glued space geodesic? Any hint ? For notation and details one can see ...
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26 views

Find a line with measure 0

A finite measure $m$ is defined on a $k$-connected set $D$, with $k>1$. You want to convert $D$ into a $(k-1)$-connected set without hurting the measure. Formally: prove that there is a set ...
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1answer
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How to prove spaces are homeomorphic using normalisation method?

Prove that the punctured plane, $X= \mathbb{R}^2\setminus \{0\}$, is homeomorphic to the complement of the unit disc, $Z=\{\underline{x} \in \mathbb{R}^2 \mid \ \|\underline{x} \| > ...
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1answer
39 views

Discontinuity of the identity function in topology

According to a theorem I was taught, the identity function $id(x)=x$ from $(\mathbb{R}, \tau_1)$ to $(\mathbb{R}, \tau_2)$ is continuous if $\tau_1 = \tau_2$. Are there any examples of topologies ...
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1answer
27 views

Why is $V(x)\cup(\mathbb{A}^2\setminus V(y))$ not quasi-affine?

I'm having trouble understanding the following situation. Apparently it's not difficult to see the union $V(x)\cup(\mathbb{A}^2\setminus V(y))$ is not a quasi-affine set. Everything is being done ...
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2answers
70 views

Two continuous functions with connected images

Suppose we have two continuous functions $f(x)$ and $g(x)$. Define $f$ on $[0,1]$ and $g$ on $[1,2]$, such that $f(1)=g(1)$. If we know that $\text {Im} (f(x))$ and $\text{Im} (g(x))$ are connected, ...
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1answer
29 views

Is this proof of $\overline{\cup}A_\alpha \subset \cup \overline{A_\alpha}$ valid?

This is a problem from my topology homework: Let $x \in \overline{\cup}A_\alpha$. Then for every $U(x)$, $U(x) \cap (\cup A_\alpha) \neq \emptyset \rightarrow U(x) \cap A_\alpha \neq \emptyset$ ...
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1answer
36 views

Are there actually 4 types of Product Topologies?

We learned about the Product Topology recently in class. So for $X,Y$ topological spaces, then we define $X\times{Y}$ to be the product topology. From what I understand, for $U\subset{X}$ and ...
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2answers
90 views

Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
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2answers
89 views

Homeomorphism between plane with different topologies

How would you show that spaces $(\mathbb{R^2},\cal{T}_r)$ and $(\mathbb{R}^2,\cal{T}_b)$, where $\cal{T}_r$ is a topology generated by jungle river metric (here) and equivalently $\cal{T}_b$ is ...
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1answer
42 views

Hilbert Subspaces: ONB

This might be a duplicate. If so, then please let me know. Thanks! Given a Hilbert space $\mathcal{H}$. Consider a dense subspace $\overline{Z}=\mathcal{H}$. Then it provides an ONB: ...
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4answers
56 views

Subset of $\mathbb{R}$ is countable iff it contains no interval?

I was doing some topology homework and used this equivalence to complete a problem, only I'm not completely sure it actually holds. My intuition tells me it's true, but I can't prove it. If it helps, ...
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0answers
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Patition of unity by real valued continuous function on compact Hausdorff space.

Let $X$ be a compact Hausdorff space and let $\{U_\alpha\}_{a \in A }$ be an open cover of $X$. Show that there exit a finite number of continuous real valued functions $h_1\cdots h_n$ on $X$ with the ...
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1answer
30 views

Any open set shares boundary with a discrete set

Claim: Let $X$ be a metric space and let $U\subset X$ be open. Then there exists a discrete set $A\subset X$ such that $\partial A = \partial U$. Approach thus far: Since this statement is about ...
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5answers
41 views

Well-ordered set with greatest element is compact

Let $X$ be a well-ordered set with a greatest element $\alpha$. We consider all sets of the form $]x,y]$ where $y \in X$ and $x$ is either another element of $X$ or the symbol $\leftarrow$ (the ...
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37 views

The Complex projective space is homeomorphic to the n-sphere

Ok I have been asked to give as detailed a proof as I can for the following question. Prove that $ \mathbb C\mathbb P^n $ is homeomorphic to $ S^{2n+1} /\sim. $ where for $ z,w \in S^{2n+1} \subset ...
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1answer
28 views

proving that a set is closed

Let X be compact, metric, connected and locally connected space. And let M$\subseteq$X closed and connected. a,b,p$\in$M are non-cut points of X. I showed that M$\setminus${p} is connected. Now I ...