# 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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### Anti-compact space

A space $X$ is called anti-compact if any compact subset of $X$ is finite. It is known that uncountable space with co-countable topology is an example of a $T_1$ anti-compact space that is not ...
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### $f :\mathbb{R}\to \mathbb{R}$ with closed fibers sending connected to connected

Let $f:\mathbb{R}\to \mathbb{R}$ be a map with closed fibers that sends connected sets to connected sets. Is it true that $f$ is continuous?
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### The map from the open disc to $\mathbb R^n$ is injective.

To show that the open disc is homeomorphic to $\mathbb R^n$, the map usually used is $f:\mathbb R^n\to D^n$ such that $f(x)=\frac{1}{1+|x|}x$. The step of showing that $f$ is injective is somewhat ...
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### Is it true that $S^n$ and $SO(n+1,\mathbb R)$ are homeomorphic ?

Is it true that $S^n$ and $SO(n+1,\mathbb R)$ are homeomorphic ? (Where $S^n:=\{x \in \mathbb R^{n+1} : ||x||=1\}$ and $SO(n+1,\mathbb R):=\{ A \in M_{n+1}(\mathbb R) : AA^t=I , \det(A)=1 \}$ )
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### For all $U\supset E$ there is a $V$ such that $E\subset U\subset V$.

$(X,T)$ a topological space and $E\subset X$. Is there a theorem that say that for all open $U\supset E$ with $U\neq E$, there exist an open $E\subset V\subset U$ with $E\neq V\neq U$ ?
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### Why is the preimage of zero of a global section of a locally constant sheaf closed?

I'm reading Faisceaux Algébriques Cohérents (Serre), no.36 lemma 2. The proof of that lemma states that the preimage of zero of a global section of a sheaf is closed when the sheaf is locally ...
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### Connectedness of a hyperbola to the x-axis

Given my definition of connected as: M is connected if it contains no proper clopen subsets. And the set H as {(x,y) : xy = 1 and x,y>0 }, with the set X representing the x-axis, is the set S = X U H ...
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### A metric space is compact iff it is pseudocompact

I was recently presented this problem from a course on topology half of which I could work out but the other half is a mystery: Let $(X, \tau)$ be a metrizable topological space, we say that a ...
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### Open and closed set in n by n matrices space. [on hold]

This problem seems to be easy yet I have no idea to deal with. If $\mathbb{M} ^n$ is the set of all real square matrices of order n, identified here with $\mathbb{R} ^{n^2}$ equipped with its usual ...
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### Is cantor set open when intersecting a closed interval $[0,\frac{1}{3}]$ (in cantor set)？

In our class, our professor said cantor set is clopen (both closed and open). One argument is that the interior of a cantor set intersecting with a closed interval, say $A = \Delta \cap [0,1/3]$ whose ...
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### Point Set Topology [closed]

For each subset B of Y, The pre-image of the interior of B is a subset of the interior of the pre-image of B. This question, for some reason, I cant seem to get the the hang of. I have tried proof by ...
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### What is a norm topology in functional analysis?

I am currently reading up about norm topology, I have a background in functional analysis but I do not know anything about topology, aside from that topology is a collection of open sets with some ...
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### What is the “Tychonoffication”?

In this link: http://mathoverflow.net/questions/23940/why-free-topological-groups-on-tychonoff-spaces I read the following: Let $X$ be a topological space. The Tychonoffication $Y$ of $X$ is the ...
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### Prove that if $F$ is finite subset of $\mathbb R,$ then $\mathbb R-F$ is an open set. [closed]

Can anyone help me to prove this problem? Definition of open set is: A subset $U$ of $\mathbb R$ is called an open set if $U=\varnothing$ or if for each $x\in U$ there is an open interval $I$ such ...
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### Why is sphere with one cross-cap homeomorphic to real projective plane?

From wikipedia Cross-cap, "A cross-cap that has been closed up by gluing a disc to its boundary is a model of the real projective plane P2 (again with an interval of self-intersection, and two ...
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### Definition of $\Delta$-complex: restriction to interior is homeomorphism

From Hatcher's Algebraic Topology, the definition of a $\Delta$-complex is: Then, a page or so later, he writes: Regarding the sentence "Condition (iii) then implies...", I don't understand why ...
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### Computing interior of subset of bounded functions

Let $L$ be the subset of those bounded functions $f$: $[-1, 1]\longrightarrow\mathbb{R}$, for which $\lim\limits_{x\longrightarrow 0} f(x)$ exists. By the iterated limits theorem, this is a closed ...
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### Show that $S^n$ is homeomorphic to $(D^n\times \{-1, 1\})/{\sim}$, where $(x, 1)\sim (x, -1)$ for all $x$ in boundary of $D^n$

I understand some basic examples of homeomorphisms such as Show that $\mathbb{R}^2/{\sim}$ is homeomorphic to the sphere $S^2$. but I don't know where to start with this example. Do I have to ...
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### How to show that absolute convergence is equivalent to permutable convergence.

I'm having difficulty understanding what this proposition is stating. I'm to show in $\Bbb R^n$ that absolute convergence of the series is equivalent to permutable convergence. I've searched my ...
Consider the following problem. If I have Banach spaces $X,Y$, and an operator $T\in B(X,Y)$, how would I show that, "If $T\in B(X,Y)$, and $p\in\overline{ T(B_{X,1}(0))}$ then $-p\in\overline{ ... 2answers 57 views +50 ### Penrose Triangle and Umbilic Torus Here are two images - one of each. It seems to me that they are the same object from the topological perspective, that one is just a smoothed-out version of the other. I think this because it is clear ... 1answer 13 views ### Calculating the Lyapunov exponent of the times-m map,$E_{m}$. I'm trying to compute the Lyapunov exponents for$E_{m}$, where$E_{m}:S^{1}\to S^{1}$,$x\mapsto mx\mod 1$. The Lyapunov exponent is given by ... 2answers 28 views ### Weak topology and the topology of pointwise convergence I'm reading the definition of weak topology in Banach Algebra Techniques in Operator Theory by Douglas: According to an article about the product topology in Wikipedia, the product topology is also ... 0answers 51 views ### Find a metric on$\mathbb{R}$with the property that the sequence of natural numbers is Cauchy. I'm trying to solve the following question: Find a metric$d$on$\mathbb{R}$that is equivalent to the usual metric and has the property that the sequence$(n)_{n=1}^{\infty}$is a Cauchy sequence. ... 1answer 56 views ### Do we need finite intersections of open sets to be open? How important/indispensable for the general theory of topology is the requirement that the open sets are closed under finite intersections? If I am not overlooking something crucial, then a huge part ... 1answer 6 views ### Image of a product of opens This is a general topology question. Let$k < n$be positive integers. Suppose we have opens$U \subset \mathbf R^k$and$V \subset \mathbf R^{n-k}$and a continuous and injective map$$f: U ... 0answers 23 views ### Is beeing open a requirement in both of these implications? I am able to show these two implications: 1. Let$X$be a topological space,$V \subset U\subset X$, U open. If V is connected in U then V is connected in X. proof: If V was not connected in X, ... 0answers 18 views ### Link of a face in an abstract simplicial complex I don't really understand this definition (from here): Ok, so we're considering a face$v$(as$K$is a simplicial complex, vertices are faces). I get that the vertex set of$\text{lk}(v)$is the ... 2answers 40 views ### All about frontier (boundary) in topological space In Engelking (theorem 1.3.2 p.24) and Choquet (prop 6.6 p.17 and exercice 17 p.112), there are several facts about frontier (or boundary) in topological spaces, but I can't find counter-examples ... 1answer 40 views ### Can a topology be recovered from its collection of pre-compact sets? Say we are given a set$X$along with a collection$\mathcal{A} \subseteq 2^X$of subsets. Is there a topology$\tau$whose pre-compact sets are exactly$\mathcal{A}$? Is that topology unique? 1answer 172 views ### A locally injection is an injection? Let$X$be a (path-)connected space, and$f:X \rightarrow Y$is a continuous mapping, if for any$x\in X$, there exist a neighbourhood$U$of$x$, such that$f$is an injection on$U$, then$f$is an ... 1answer 28 views ### Hausdorff and compactness I am curious about this problem: Let$X$be a topological space. If every$Y\subseteq X$is a discrete space whenever$Y$is compact, then is it true that$X$is Hausdorff? My attempt: let$x,y\in ...
I am having difficulty with a problem in one of my textbooks. It gives three definitions $1$. A set is called a closed set if its complement is open. $2$.The closure of a set E is the intersection ...