# Tagged Questions

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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### closure of inverse image is subset of inverse image of closure, given that $f$ is continuous

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. Prove then that $$\overline{f^{-1}(X)} \subset f^{-1} (\overline{X})$$ for every $X \subset \mathbb{R}$. Attempt at proof: Let ...
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### Symmetric bilinear forms and (continuous) dual spaces

Let $V$ be an infinite dimensional locally compact vector space over a field $k$ (the field $k$ has the discrete topology and on $V$ we fix the linear topology ). Moreover suppose that on $V$ is ...
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### Show that $X_K$ is connected.

Let $\{X_\alpha\}_{\alpha}$ be a collection of connected spaces.Let $X$ be the product space $X=\prod_\alpha X_\alpha$ . Let a=$(a_\alpha)$ be a fixed point of $X$. a.Given any finite subset $K$ ...
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### How to prove this topology equality?

Suppose $(A,\tau_A)$ is the subspace of $(X,\tau)$, show that for all $B\in 2^A$ the following relationship holds: $$\text{int}B=\text{int}_A B\cap \text{int} A.$$ Here subtopology $\tau_A$ is defined ...
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### Two neighborhoods of $0$ in the plane and the upper half plane (resp.) cannot be diffeomorphic. [duplicate]

Let $\mathbb{R^2}$ be equipped with the standard topology, and let $\mathbb{H^2}$ be the upper half plane (containing the x-axis), equipped with the subspace topology. Let $U$ be an open neighborhood ...
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### Does $\overline{ B_r(p) }=B_r[p]$ imply that (X,d) is connected?

Let's consider a metric space with it's typical base. Suppose that $\overline{ B_r(p) }=B_r[p]$. It implies that the metric is continuous, if we consider space (X,d) with X as domain and ...
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### Existence of subdivision of PL manifold triangulation which is combinatorial manifold

Suppose $X$ is a PL manifold with triangulation $\psi:|\Delta| \to X$. Does there exist a subdivision of $\psi$ which is a combinatorial triangulation?
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### Prerequisites for Kirby Calculus?

I've looked around, but I haven't found anything in particular on Google or here, so I figure I'd ask. What are some solid prerequisites to be able to tackle Kirby Calculus? I have a solid ...
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### Why would the category of topological spaces be a balanced category (i.e. monic epimorphisms are isomorphisms)?

I've just read on this page that For example, $\mathsf {Set}$ (the cateogry of sets), $\mathsf {Grp}$ (the category of groups), and $\mathsf {Top}$ (the category of topological spaces) are all ...
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### Does this power law for cartesian products hold and does it come from a homeomorphism?

Feel free to edit the title of the post or leave a comment if you think that the stated problem is not accurately reflected by the title. Let $(X_i, \tau_i)_{i \in I}$ be a family of topological ...
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### Infinite intersection of compact, path connected, nested sets is path connected?

I showed that given $A_1\supseteq A_2, ...$ compact, connected sets, $\bigcap_{i=1}^n A_n$ is connected but is the statement true if we replace connected with path connected? Is there an ...
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### How to compare open sets from $\mathbb{R}$ with the finite complement topology and the standard topology?

I am being asked about which of the two topologies are finer than the other and I am having troubles once I am not able to say what are a basis for $\mathbb{R}$ with the finite complement topology, so ...
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### If a measure only assumes values 0 or 1, is it a Dirac's delta?

Let $\mu$ be a probability measure on a metric space $M$ (with the Borel $\sigma$-algebra). If $\mu(A)\in \{0,1\}$ for all measurable set $A\subset M$, then: Is it true that $\mu$ is a Dirac ...
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### The concept of net and subnet in topology

My question mainly lies on the concept of net and subset. I have some difficulty in understanding and using this tool. As far as I understand, the specific directed set (the neighborhood system of one ...
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### A reference for patch topology

I saw a paper that defines the definition of patch topology on the Spectrume of a commutative ring with identity. Is there any reference for this concept that give some property for this topology? ...
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### Does double pendulum pass through all the points in its reagion?

In the previous question, I asked for PDF of the locus of end of a double pendulum. Now, I am thinking about a more fundamental question. When a double pendulum moves, does its trajectory make a ...
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### Determine whether the two sheeted paraboloid $x^2-y^2-z^2=1$ is connected

It is disconnected right? Just looking at the graph, there is a gap between the two sheets so it should be easy to write the set of points on the surface as the union of two open, disjoint, non-empty ...
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### Diagonalizable matrices

Question is to prove that the set of all diagonalizable matrices are dense in $M_n(\mathbb{C})$. I am sure this question is discussed in this site previously but i am looking for a more constructive ...
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### can a sequence of distinct integers be convergent? Rudin

Let {Xn} be a sequence of distinct integers. Can it be convergent? This is from Rudin, and I assumed that {Xn} is given to be an infinite sequence since it's defined as a function which maps from N. ...
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### Homeomorphisms between any two doubly punctured spheres and two punctured $R^n$.

Let $p, q$ be the north pole and the south pole of $S^n$ respectively. Then $S^n-p-q$ is homeomorphic to $S^n-a-b$ where $a,b$, are distinct points in $S^n$. Also $R^n-a$ is homeomorphic to ...
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### Gluing two solid tori by a homeomorphism of their boundarries.

I am aware that we get all lens spaces by gluing two solid tori by their boundaries. My question is, do we get more spaces besides lens spaces? in other words, do all homeomorphisms of the boundaries ...
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### Example of a continuous surjection from $I=[a,b], a,b \in \mathbb{R}$ to $S^2$.

Continuous surjection from $I=[a,b], a,b \in \mathbb{R}$ to $S^2$. What is an example of such a surjection? I can't think of any. I would greatly appreciate any examples.
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### Is every Closed set a Perfect set?

From 'baby' Rudin. I've seen that a set is closed iff it contains all of its limit points. In Rudin, $(d)$ says if every limit point of E is a point of E, then $E$ is closed. He also says $(h)$: $E$ ...
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### Curvature and topology

I am studying Riemannian Geometry and I came across various Theorems which give conditions on the topology of a manifold given conditions on curvature, and vice-versa. Just to mention a few of them: ...
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### Books and sources concerning $G$-spaces

A $G$-space is (generally) a topological space $X$ equipped with a continuous action by a topological group $G$. I mean generally because, I've never studied before $G$-spaces and after I read a ...
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### When does contractible space of almost complex structures taming a given symplectic form $\omega$ contain an integrable compatible one?

Given a symplectic form $\omega$ on a compact symplectic manifold $X$, we know there is a contractible homotopy class $\mathcal{J}_{\omega}$ of almost complex structures that tame $\omega$. A subset ...
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### A function $f:X \rightarrow Y$ and $A$ is a subspace of $X$ such that $f|A$ is continuous but $f$ is not continuous at any point of $A$.

I am not able to find any example. Any hint will be appreciated.
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### If $A\subset \Bbb R^n$ is a set such that for every finite, nonempty $B\subseteq A$, $A\setminus B$ is closed, show that $A$ is closed.

Let $A\subseteq \Bbb R^n$ be a set such that for every finite, nonempty $B\subseteq A$, we have that $A\setminus B$ is closed. Is $A$ is closed? I already proved that every point of $A$ is an ...
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### Klein bottle from mobius strip

We are trying to make a klein bottle using 2 mobius strips on Mathematica. We have the mobius strip parametrization equations and then we inversed the equations to get a reflected mobius strip. Now ...
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### Homotopic maps have homotopy equivalent mapping cones

Let $f,g:X\to Y$ be maps of spaces such that $f\simeq g$. Is it true that the mapping cones $\operatorname{cone}(f)$ and $\operatorname{cone}(g)$ are homotopy equivalent? Can we write down an ...
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### Subspace topology vs order topology in subspaces

For a space $X$ and a linear order $<$ on $X$, we can define a topology $\tau$ by the basic open sets $U_{a,b}=\{x\in X: a<x<b\}$. Now, let $A\subset X$. We can endow it with the subspace ...
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### Literature on proof that (complex) Stiefel manifold is a compact topological manifold?

I am searching for literature for a basic proof that the (complex) Stiefel manifold is in fact a compact topological manifold but a i cant find any without the use of Lie-groups and such. Can anyone ...
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### closed union of closed sets

The following is well-known: if $X$ is a topological spaces, then the union of compact subsets in it need not be compact. But, if $I$ is a compact set in the hyperspace $H(X)$ of all compact subsets ...
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### Closed continuous operator has closed domain? Question about completeness

This question is about the statement: Let $X$, $Y$ be normed linear spaces and $D$ a linear subspace of $X$ and suppose that $A\colon D \to Y$ is a linear operator. If $A$ is continuous and closed ...
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### How are related Connected Component with Open-Closed Subset

I used to think that connected components were closed and open at the same time, but I discovered quite recently (italian page of Wikipedia) that this might not be the case. What I'm asking for is: ...
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### Generalization of cantors intersection theorem

Let $A_1\supset A_2\supset\cdots$ be a sequence of connected compact subsets of $\mathbb{R}^2$. Is it true that their intersection $A=\bigcap_{i=1}^{\infty}A_i$ is connected also? Suppose it is not ...
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### Describing this topological space

I am doing this exercise from Armstrong's Topology book, and am super confused: "describe each of the following spaces: [...] $\mathbb{E}^2$ with each of the circles centre the origin and of ...
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### show that $x^4+y^4+z^4=1$ is compact and has no boundary

I'm trying to show $x^4+y^4+z^4=1$ is closed surface. so i finally reached to this definition: a closed surface is compact and has no boundary. despite i had learned several elementary topological ...
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### Questions on set theory, topological spaces, and continuity.

(My question would not have fit in the title) Questions: a) Let X be a euclidean three dimensional space R$^3$ with the standard Euclidean length and let Y $=\{P_1$such that $P_1=(a_1,b_1,c_1)$and ...
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### Why subspace of a compact space not compact

In topology, the compact set conveys the idea that all points are not too "separated", since how separated is determined by the topology of the space, so we can roughly say for a compact space, each ...
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### Connected components and continuous functions

This may be a stupid question. If $X$ is a topological space, I know that being a connected subset of $X$ is not necessarily equivalent to being a path connected subset of $X$. But does it hold that ...
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### Distance between compact sets

Let $K$ and $L$ be nonempty compact sets, and define $$d = inf\{\lvert x-y \rvert : x \in K, y \in L\}$$ If $K$ and $L$ are disjoint, show $d \gt 0$ and that $d = \lvert x_{0}-y_{0} \rvert$ for ...
### Prove that if C is a closed set and $L = sup (C)$, then $L \in C$.
Prove that if C is a closed set and $L = sup (C)$, then $L \in C$ This is what I have in mind, but I am not sure if it is quite right. Proof: Assume $L \notin C$, then $L \in C'$. Since C is ...