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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Question regarding notation in algebraic topology

My class has not been following a book and my professor's last bit of notation is a bit confusing to me. This is the goal. We are given a path-connected space $Y$ and $H$ a subgroup of $\pi_1(Y,y)$. ...
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3answers
25 views

Seeming ambiguity in the definition of open sets?

Concerning open sets: A set $S$ is "open" if and only if it is a neighborhood of each of its points. But for $S$ to be a neighbourhood of its points if there is some other set $V$ which contains an ...
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1answer
17 views

Why existence of universal covering implies that the base space be locally path connected?

I am reading Chapter 13, the chapter about classification of covering spaces, of J.Munkres' Topology. My confusion raised when I read Corollary 82.2. which says: the space $B$ has a universal ...
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0answers
26 views

$A$ is an interval so $A$ is connected?

I want to prove that if $A\subset \mathbb{R}$ is an interval then $A$ is connected. I found this proof, and I don't understand it essentially the ii) Suppose that $A$ is an interval but not ...
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0answers
15 views

How do you show Euler characteristic of any convex polyhedron is $2$?

In the Euler characteristic proof of a convex polyhedron, how do you show the cell decomposition of projection of two polyhedra 1) have a common refinement AND 2) that common refinement comes from ...
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1answer
52 views

Is this a topology?

Suppose that we have a set $S$ containing 0 and 1. Can we define our topology to be the four open sets $\varnothing$, $\{0\}$, $\{1\}$ and $\{0,1\}$? I know that the Sierpinski set contains the three ...
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1answer
24 views

Is $A$ compact, $f(A)$ uniformly continuous and is $f^{-1}$ continuous?

$X$ and $Y$ are metric spaces, $A\subseteq X$, $A$ is bounded. map $f:X\to Y$ is continuous. Questions: Is $A$ necessarily compact? Is $f(A)$ uniformly continuous? If given that $f$ is a ...
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43 views

Is $\mathbb{R}^n$ not nowhere dense?

Is $\mathbb{R}^n$ not nowhere dense? I am trying to show that a clopen set is not necessarily nowhere dense. I know that it is both open and closed, but I am not sure how to find the interior of ...
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24 views

Borel Measures: Atoms (Summary)

Disclaimer: This is a summary of the discussions: Measure Atoms: Definition? Borel Measures: Discrete & Continuous? Borel Measures: Atoms vs. Point Masses Reference: Further results are ...
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1answer
19 views

On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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1answer
40 views

Find a topological space X and a compact subset A in X such that closure of A is not compact.

Find a topological space X and a compact subset A in X such that closure of A is not compact. I first concluded that we must have X to be a non compact and a non Hausdorff space so that closure of A ...
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1answer
29 views

What does “the support of $f$ lies in $V$ mean?”

I have come across similar phrases and I am not sure what they mean. For example, if the phrase states "the support of $f(x)$ lies in a set $V$, does it mean that $V$ contains all $x$ such that ...
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9 views

$\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
19 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 compact, then $Y$ is locally compact. I wonder, does this hold for ...
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2answers
22 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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3answers
26 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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1answer
28 views

Zariski-open subset in $\mathbb{C}^n$ to Zariski-closed subset in $\mathbb{C}^{n+1}$

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
39 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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0answers
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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15 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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2answers
75 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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0answers
35 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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2answers
279 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 ...
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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 ...
2
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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
29 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
19 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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1answer
45 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
44 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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0answers
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
12 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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0answers
22 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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0answers
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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0answers
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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0answers
43 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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0answers
36 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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0answers
20 views

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
24 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
19 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
27 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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0answers
32 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
32 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 $h$ ...