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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$\overline{M\setminus X}=M\setminus\operatorname{int}(X) $? [duplicate]

Why is true that for $X\subset M$ (where $M$ is a metric space), we have that $\overline{M\setminus X}=M\setminus\operatorname{int}(X) $? Not sure why.
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0answers
17 views

Is local compactness preserved by continuous closed onto functions? [duplicate]

I've just shown for a homework problem that if $f$ is an open continuous function from $X$ onto a $T_2$-space $Y$, and $X$ is locally connected, then $Y$ is locally compact. I wonder, does this hold ...
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2answers
19 views

If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable

I need to show that: If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. I have already showed that every locally compact Hausdorff space ...
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2answers
23 views

Prove the set, {y ∈ X | r ≤ d(x,y) ≤ s}, is closed

Let r < s be positive real numbers and x ∈ X. Prove that the set: {y ∈ X | r ≤ d(x,y) ≤ s}, is closed. Having trouble with how I should tackle this ...
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2answers
16 views

Quick question: functions to spaces with equivalence relations

So I'm a little confused about sending functions from spaces without equivalence relations to a space with equivalence relations. For example, I'm trying to define a function $f : S^{n} \rightarrow ...
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18 views

Zariski-open subsace

Let $n \in \mathbb{N}$, $f\in \mathbb{C}[X_1,\dots,X_n]$, and $D(f):=\{x=(x_1,\dots,x_n)\in \mathbb{C^n}|f(x)\neq 0\}.$ I want to show that there is a injective $\Phi: D(f) \rightarrow ...
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1answer
37 views

What kind of Choice am I making in this argument?

I have an argument that's supposed to imply Choice, but I'm afraid it may be using some choice. If it does, how much choice? This is the part of the argument that might use some Choice. I marked the ...
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21 views

Power sets and Discrete Topologies.

These are the definitions I have learnt; Let $X$ be any non-empty set and $\tau$ be the collection of all subsets of $X$. Then $\tau$ is called the discrete topology on the set $X$. The ...
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13 views

Cofinite Topology: Borel Algebra?

Given the cofinite topology: $$\mathcal{T}:=\{U\subseteq\Omega:\#U^c<\infty\}$$ and generate its Borel algebra: $$\sigma(\mathcal{T})=\{E\subseteq\Omega:\#E\leq\aleph_0\lor\#E^c\leq\aleph_0\}$$ Why ...
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64 views

Borel Measures: Atoms vs. Point Masses

Let a measure be $\mu:\Sigma\to\mathbb{R}_+$. Call a measurable $A\in\Sigma$ an atom if: $$\mu(A)>0:\quad\mu(E)<\mu(A)\implies\mu(E)=0\quad(E\subseteq A)$$ and a singleton $\{a\}\in\Sigma$ a ...
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31 views

Retraction to an interval in a metric space

Suppose that $X$ is a metric space and $A$ is a subspace of $X$ that is homeomorphic to the interval $[0,1]$ with its usual topology. Let $v$ and end point of A. How do you proof that there is a ...
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274 views

Is continuity in topology well-defined?

In topology, a function is continuous if inverse of every open set is open. But for the inverse to be well-defined the function should be bijective. For example consider the projection map. It is not ...
3
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2answers
26 views

Applications of Baire's Threom [duplicate]

In a lecture on Baire's Theorem (for complete metric spaces), I gave, for a rather advanced undergraduate class in Real Analysis (covering the theory of metric spaces and elements of general ...
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4answers
29 views

Union of infinite many closed sets

If $(K_i)_{i \in \mathbb{N}}$ is a sequence of closed sets in $\mathbb{R}^3$, then the union of these sets $\bigcup_{i=1}^\infty K_i = K_1 \cup K_2 \cup ... $ is also closed. My idea: ...
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1answer
24 views

Is this enough to prove a homeomorphism? — inverse on a dense subset

I want to prove that a map $f:A\to B$ is a homeomorphism, I know that $A$ is compact. I am not sure whether it is enough to show that: $f$ is continuous and injective for all $y\in B_1$, there is a ...
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1answer
27 views

Cantor's intersection Theorem without the diameter hypothesis

In proving Cantor's in intersection theorem, the fact that limit of the diameter of the sets is 0 was used to prove that the intersection is non-empty. I just wondered if that hypothesis is excluded ...
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2answers
27 views

Determining whether a set is open and bounded

I know that given $a < b$ and $g(x) \le h(x)$ $\{(x,y) \in \mathbb{R}^n |\ a \le x \le b, \ g(x) \le y \le h(x) \}$ is a closed constrained/bounded/limited (not sure what the terminology is in ...
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1answer
15 views

Euler characteristic of a convex polyhedron

In the Euler characteristic proof of a convex polyhedron, how can you show two cellular decompositions of two different polyhedron contain a common refinement?
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0answers
35 views

Cutting a torus enough times disconnects it

I am interested in showing that if you cut a torus too many times it becomes disconnected. Let $\mathbb T^n$ be the standard $n$-dimensional flat torus. Let $M_1, \ldots, M_k$ be $k$ disjoint smooth ...
4
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1answer
41 views

Universality with respect to quotients

Is there an infinite cardinal $\kappa$ for which the following statement (S) true? (S) : There is a topology $\tau_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq ...
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36 views

Question about topological properties of $\Bbb{C}_p$

It is known that the structure of $p$-adic integers, $\Bbb{Z}_p$ is homeomorphic to the Cantor set, and $\Bbb{Q}_p$ is homeomorphic to the one-point deleted Cantor set (as I know, I don't certain it.) ...
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0answers
11 views

Basic Topology: Armstrong

I am currently reading basic topology by Armstrong and he references "thickening" a tree. I am not sure what this means. Can anyone briefly explain?
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1answer
14 views

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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21 views

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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33 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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48 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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1answer
18 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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42 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
43 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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35 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
23 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
11 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
18 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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1answer
21 views

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
26 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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31 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
31 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
42 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
186 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
20 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
26 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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1answer
39 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
43 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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251 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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2answers
48 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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2answers
25 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
40 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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2answers
31 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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2answers
39 views

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 $d(x,F) = \inf d(x,y)$ ...