Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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2answers
29 views

What does “finitely many” mean here? “$B$ is a collection of sets that are the intersection of finitely many sets in [a set of sets]”

Take a set of sets $C$. What does it mean when we say "$B$ is a collection of sets that are the intersection of finitely many sets in $C$?" Does "finitely many" mean "from 1 to n"? With $1$, I ...
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1answer
52 views

If a topology over an infinite set contains all finite subsets then is it necessarily the discrete topology?

If we define a topology via open sets then yes, infinite unions of the finite subsets will necessitate the infinite subsets also part of the topology giving us the discrete space; however, if we ...
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1answer
281 views

Nets, dense subsets and continuous maps

Let $X$ and $Y$ be topological spaces, with $Y$ regular. Consider a dense subset $D\subset X$, a continuous map $f:D\rightarrow Y$, and a map $g:X\rightarrow Y$ (i.e. $g$ is not assumed continuous). ...
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3answers
46 views

Continuity in a topological group

The group of the real numbers under addition is a topological group ($\mathbb{R}$,$+$) with the usual topology. However, I can't see why the group operation (addition) is a continuous function. Would ...
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1answer
156 views

The closed points of a scheme and irreducibility

Suppose $X$ is a scheme. We say a point $x$ is closed in $X$ if $\overline{\{x\}}=\{x\}$. Let $t(X)$ be the subspace of all closed points in $X$. We say X is irreducible if its topological space is ...
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1answer
55 views

Given a function $f$ infinitely differentiable at a point $c$ does there exist a neighborhood of $c$ in which $f$ is infinitely differentiable?

Suppose $f$ is a real valued function defined on a subset of the reals and $f$ is infinitely differentiable at $c$.Then is it possible that there does not exist any neighborhood of $c$ in which $f$ is ...
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1answer
18 views

Locally compact space versus locally connected space

I remember when I was studying the notion of local connectedness it meant that each point has "arbitrarily small" neighborhoods that are connected. More precisely, one has the following definition: ...
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0answers
20 views

Is SO(n) a deformation retract of SO(n+1) for $n\ge 3$?

I know that $SO(n)$ shares many topological invariants with $SO(n+1)$ for $n\ge 3$ (such as first fundamental group, first singular cohomology class over $\mathbb{Z}$, etc.). I also know that there is ...
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0answers
17 views

Continuous functions on $J\times J\rightarrow \mathbb{R} $ can be uniformly approached by continuous functions: $f_1(x) g_1(y)+…+f_n(x) g_n(y)$

Let $J\subseteq \mathbb{R}$ be a compact interval and let $\mathbb{A}$ be a collection of continuous functions on $J \rightarrow \mathbb{R}$ which satisfy the properties of the Stone-Weierstrass ...
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1answer
13 views

topology - stability under intersection

I am facing the following definition of topology: A topology on a set $X$ is defined as a subset $\cal O$ of the power set $\cal P$ of $X$, that: contains the empty set and $X$ is stable over the ...
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2answers
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Have I shown that convex subsets are connected with correct generality? Any flaws in my proof? (Baby Rudin 2.21(C))

$\exists a_1, b \in C_1 \subseteq R^k$ such that there is another pair of points $a_2, b \in C_2 \subseteq R^k$ where $C_1$ and $C_2$ are convex. A convex subset $C$ is defined pairwise by the ...
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1answer
18 views

Distribution of areas of random slices through a cube

As an applied mineralogist I am always dealing with the stereology of taking measurements on a 2D slice through aggregates of multiple 3D objects, i.e. rock. If I assume the 3D object is a sphere, the ...
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1answer
11 views

Boundedness of $f^{-1}(B)$ for some bounded set $B\in X$

If $f$ is a continuous mapping from a topological space $X$ onto itself, then is there any example of $f$ such that $f^{-1}(B)$ is not bounded for a bounded set $B\in X$?
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2answers
22 views

Why $\{(x,y)\in\Bbb{R}^2 :\, y=x\sin \frac1x\}\cup\{(0,0)\}$ is connected but not compact?

I wonder why the set of all points in the plane satisfying $y = x\sin \frac {1}{ x}$ together with the origin is connected but not compact. Is there any example of a open cover that is not finite?
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1answer
19 views

Compatible atlas induced the same topology.

Let $M$ a set and $\mathcal{A}=\{(U_\alpha,\varphi_\alpha)\}$ an atlas we said that $A\subseteq M$ is open iif $\varphi_\alpha(A\cap U_\alpha)$ is open in $\mathbb{R}^n$ for all chart $(U_\alpha,\...
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1answer
13 views

Show every continuous function on [0,$\pi$] is the uniform limit of sequences of function of “polynomials of sin(kx)”(Stone Weierstrass theorem)

The exercise is 26E on Bartle´s Elements of Real Analysis. It asks to use the fact that every continuous real valued function on $[0,\pi]$ is the uniform limit of a sequence of functions of the form: ...
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0answers
36 views

About Bernstein sets

Remember that a subset $X\subseteq \mathbb{R}$ is a $G_{\delta}$-set if $X$ is a countable intersection of open sets in $\mathbb{R}$. For example closed subsets of $\mathbb{R}$ are $G_{\delta}$-sets. ...
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0answers
25 views

Prove the existence of a certain point of convex set

Let $S$ be a closed convex set and let $x \notin S$. Prove that there is only one point $y \in S$ such that $$|x-y| \le |x-z|, \quad \forall z\in S$$
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How to compute the integral over a parametrized Torus ? $\int\int_T (x-2)^2dS$?

The question I need to solve is: T is a Torus obtained by rotating the center circle (2,4,2) with radius 2 contained in the z = 2 plane around the line {y = -1, z = 2}. I know I need to parameterize ...
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1answer
22 views

Double derived set in $T_0$ spaces

Let $A$ be a subset of a topological space $X$. I am interested in establishing under which conditions the following inclusion holds: $A'' \subseteq A'.$ This is certainly false in general: consider ...
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2answers
32 views

Incomplete Metric Space in R - Examples

Which sequence in $\mathbb{R}$ do not converge with the metric d(x,y)=|$\frac{x}{1+|x|}-\frac{y}{1+|y|}$|? I am struggling to think of a sequence to get the numerator to increase faster than the ...
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Explanation of Blaschke's selection theorem

Here's the first part of the proof of Blaschke's selection theorem. I have two questions that I couldn't figure out: What is the base case $m=1$? For the underlined part, how can we be sure that the ...
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2answers
115 views

Doubt about proof in Tube Lemma

So i have been studying topology and when proving that the finite product of compact spaces is going to be compact we have to use the tube Lemma, and we have to prove it. I have a question about the ...
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1answer
31 views

How to show this function $h:\Bbb{R}^n\setminus{\{0\}} \to S^{n-1} \times \Bbb{R}$ is continuous?

Is the function $h:\Bbb{R}^n\setminus{\{0\}} \to S^{n-1} \times \Bbb{R}$ defined by $x=(x_1, \dots, x_n) \mapsto \left(\frac{x_1}{\|x\|}, \dots, \frac{x_n}{\|x\|}, \log\|x\|\right)$ continuous? I ...
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1answer
49 views

Proof that S^1 is contractible?

I couldn't quite understand why $S^1$ shouldn't be contractible into one point. I came up with the following homotopy which seems to show that $S^1$ actually is contractible: Let $x \in S^1$ and $H: ...
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2answers
17 views

A simply question on open set.

Let $\mathbb{R}^n$ with the usual topology. We suppose that $A, B\subseteq \mathbb{R}^n$ are open, and suppose we have shown that $$A=B\cap C,$$ where $C\subseteq\mathbb{R}^n$ Question. Can I ...
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0answers
29 views

Topology: Homeomorphism from R^2 to itself preserving standard topology

I need to prove that there exists a homomorphism f from R2 to itself with its standard topology such that: when considering any pair of three distinct points (x1, x2, x3) and (y1, y2, y3), f maps (...
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0answers
30 views

Topology: What Defines (non-Trivial) Paths as being the Same Trace (Curve)?

(In the absence of feedback on this site I've also posted the question at https://mathoverflow.net/q/343762) I think I know the answer but can't prove it. Assume that Y is a Hausdorff space and ...
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0answers
28 views

Question about extension of homeomorphism

Suppose $A$ is homeomorphic to a 2-sphere in 3-dimensional Euclidean space. Since $A$ is a closed surface in 3-dimensional Euclidean space, I wonder if the interior of $A$ together with $A$ itself can ...
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2answers
45 views

Find out given space is $T_{2}$ or not

Question : On usual topology $R^{1}$, $\sim$ is a equivalence relation if $x-y$ is in $\mathbb{Q}$ (the set of rational numbers), then $x\sim y$. If Quotient space $R^{1}/\sim$ has quotient topology, ...
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1answer
124 views

Is the set of convex bodies include in a closed ball compact?

I consider the set $\mathcal{K}_B$ of convex bodies (convex and compact) which are include inside the unit closed ball of $\mathbb{R}^d$. I endow this set with the Hausdorff distance. Is it compact?
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1answer
8 views

Construction of a vector valued function that maps from $\mathbb{R}^n$ to an embedded disc

Does there exist a continuous, surjective, vector valued function $f: \mathbb{R}^n \to B$ where $B$ is the disc embedded in the positive portion of $\mathbb{R}^n$ $B = \{x \in \mathbb{R}^n| x_i \...
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1answer
35 views

What does it mean to take a derivative in a space of countable elements? [on hold]

Suppose you have the space of rational numbers, $\mathbb{Q}$, with the standard metric. Now, notice how I said "space", not "subspace", which means you don't have the same topology of treating $ \...
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0answers
18 views

Baire's category Theorem examples [on hold]

What are the examples of use of Baire's Category Theorem in general topology?
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0answers
17 views

Let X=C with the usual metric and A={(x,y):y=sin1x,0<x≤1}. Show that cl(A)=A∪{(0,y):−1≤y≤1}.

Let X=$\mathbb{C}$ with the usual metric and A=$\{(x,y):y=sin\frac{1}{x},0< x \leq 1\}$. Show that cl(A)=$A∪\{(0,y):−1 \leq y \leq 1\}$. Hint: Each open ball centred at (0,y), $-1 \leq y \leq 1$ ...
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1answer
27 views

Prove that restricted quotient map is homeomorphism

Let $\mathbb{R}/\mathbb{Z}$ be the quotient of $\mathbb{R}$ by the equivalence relation $x\sim y\iff x-y\in\mathbb{Z}$, endowed with the quotient topology, and let $\pi:\mathbb{R}\to\mathbb{R}/\mathbb{...
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1answer
39 views

A proof on continuous functions

I want to prove this statement, provided it is true: "A function is continuous at $x_0$ iff for every open $O$ containing $f(x_0)$, there exists an open $U$ containing $x_0$, such that for every $x\...
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1answer
69 views

pullback diagram & connected, irreducible or reduced schemes

Let $X, Y,Z$ schemes with maps $f:X \to Z, g:Y \to Z$. we take a look at the 'pullback' diagram $\require{AMScd}$ \begin{CD} Y \times_Z X @>p_X>> X\\ @Vp_YVV @VVfV\\ Y @>g >> Z \end{...
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1answer
17 views

Quotient Topology and Identifying Points

I am working on some examples where we identify points and figure out what the quotient topology looks like. Below is the four examples with my guesses. Am I right? What do I need to change? The disk ...
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1answer
53 views

$A = \{(x, y): y = \sin \frac1x , 0 < x \leq 1\}$. Show that $\operatorname{cl}(A) = A \cup \{(0, y) : -1 \leq y \leq 1\}$.

Let $X = \Bbb{C}$ with the usual metric and $A = \{(x, y): y = \sin \frac1x , 0 < x \leq 1\}$. Show that $\operatorname{cl}(A) = A \cup \{(0, y) : -1 \leq y \leq 1\}$. $\operatorname{cl}(A)$ ...
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0answers
46 views

Reference request. Invariance of the ball by an application.

I'm writing an article and didn't want to reproduce the proof of lemma below. I just wanted to state and indicate a reference in English language to the proof of the lemma. I have a reference in ...
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2answers
23 views

Comparison between $\sup W$ and $\sup \overline{W}$

It is well-known that $\sup S= \sup \overline{S}$ for any nonempty $S\subset \mathbb{R}$. Now, let $W$ be a nonempty subset of $\mathbb{C}^d$ and $\overline{W}$ be the closure of $W$ with respect to ...
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1answer
13 views

A problem concerning a set’s closed intersections implying closedness

I have a feeling I didn’t quite get this problem from Munkres right. I’d like to proceed by contradiction, but feel like I’m missing some insight, which means I might be lying accidentally somewhere ...
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0answers
50 views

Intuition for Freudenthal Suspension

One version of the Freudenthal suspension theorem is the following: Suppose a CW complex $X$ is a union of two subcomplexes $A,B$ with $A\cap B\neq\emptyset$ connected and nonempty. If $(A,A\cap B)$...
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0answers
14 views

What does it mean to say that the space of convex bodies is locally compact with respect to the Hausdorff metric?

I know that this is equivalent to requiring each convex body in the space having a compact neighborhood. Still I can't visualize this.
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1answer
44 views

Surface with fundamental group $\pi_1(A)\cong \mathbb{Z}_2$ embeddable in $\mathbb {R}^3$?

It’s well known that the surface formed by gluing up opposite boundary points on a closed disc has fundamental group $\mathbb{Z_2}$. However, it seems that surface can’t be realized in $\mathbb{R}^3$ ...
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0answers
65 views

Variant Of Inscribed Square Problem

Looking at the inscribed square problem, I noticed that for some continuous curves $c:S^1\to\mathbf{R}^2$, one can prove that every continuous curve $c'$ with $\|c-c'\|_\infty<\epsilon$ contains an ...
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1answer
39 views

Topologies generated by arbitrary sets.

In my Point Set Topology class, we learned about topologies generated by bases and sub-bases. However, I think I have a notion of a topology generated by arbitrary subsets of the powerset of a given ...
2
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1answer
61 views

A compact subspace of metric space with no isolated points.

Suppose $X$ is a metric space which is connected and has no isolated points. Then I want to show that it will have some non-empty compact subspace $Y$ such that $Y$ has no isolated points. I don't ...