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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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Product of Mrówka space and one point compactification discrete space.

I was reading an article and I have some troubles to understand it. First, the required definition to understand the problem: Let $\mathcal{U}\subseteq \{A\subseteq\omega: |A|=\aleph_0 \}$. We say ...
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Product of compact metric spaces is again compact.

Proof Let $(X_j,d_j)$ be a compact metric space for $j=1,...,n$. Denote $X=X_1 \times X_2 \times...\times X_n$ to be the product of compact metric spaces. Assume the property that the open sets in $(...
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1answer
32 views

The only meagre and open set is the empty set?

Suppose that A is a meagre set in a topological space X. I think that the answer of my question is yes, because $\mathring {A}$ is empty. Am I right?
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2answers
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What is the topology associated with the algebras for the ultrafilter monad?

It is easy to find references stating that the category of compact Hausdorff spaces $\mathbf{CompHaus}$ is equivalent to the category of algebras for the ultrafilter monad, $\mathbf{\beta Alg}$. After ...
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2answers
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Does the definition of the set $S^n$ have implied algebraic structure?

Let's say I wish to define a topological space $(S^1, \mathscr{O})$. I would imagine that such a space is only made of two structures, namely two sets. However, the definition of $S^n$ is given by: $$...
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15 views

Meromorphic Function on a Riemann Surface

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at pages 73/74): In the proof we construct locally in $U$ a polynomial $...
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30 views

Does this condition for finding the limit points work?

I am reading Topology Without Tears, and I've been scribbling notes in the margin trying to streamline some of Morris's more... inefficient explanations. Being the soulless automaton that I am, I ...
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25 views

Closed graph and fixed points

I’m currently trying to understand the following Proposition from a paper i’m reading: Prop.: Let $X$ and $Y$ be two Hausdorff topological linear spaces. Let $H:X \times Y \rightarrow Y$ be a ...
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1answer
45 views

Connectedness of complements

Let $A$ and $B$ are compact subsets of $\mathbb{C}$ such that $B=A \cup$ iso($B$). If $A^c$ is connected then prove that $B^c$ is connected. Here iso($B$) denotes the set of isolated points of $B$. ...
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Condition for Riemannian distance to be equal to metric distance

If $M$ is a metric space than it is a topological space and if it is locally homeomorphic to to $R^n$ we say that it is a manifold and if we equipped this manifold with a inner product $g_p$ on ...
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1answer
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Prove that topological space $ X= {[0,1]}^{2} $ with dictionary order topology is not second countable. (my solution)

Prove that topological space $ X= {[0,1]}^{2} $ with dictionary order topology is not second countable. I would assume that it is second countable. So then there would exist a basis $B$ of $X$ which ...
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1answer
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On topological space $ X ={ <0,1>} ^{2}$ with dictionary order topology find the weight of the space $X$.

On topological space $ X ={ <0,1>} ^{2}$ with dictionary order topology find the weight of the space $X$. I am siimply stuck don't even know where to start.
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2answers
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Prove that the weight of standard topology cannot be $\in \mathbb{N}$.

Prove that the weight of standard topology on $\mathbb{R} $ cannot be $\in \mathbb{N}$. The weight is the minimal cardinality of all bases. It seems simple enough, I would assume it is finite and ...
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Let ($X_{i} , i \in I$) be family of disjoint topological spaces, where each of them weight $\alpha$

Let ($X_{i} , i \in I$) be family of disjoint topological spaces, where each of them weight $\alpha$ and assume that on $X = \bigcup(X_{i})_{i \in I}$ we have a weak topology. If the card($I$) $\leq \...
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1answer
51 views

Quotient to make $X$ a $T_1$ space

Let $X$ be a topological space. We define a relation on $X$: $$x \approx y : \quad \Leftrightarrow \quad x \in \overline{\{y\}}.$$ In general $\approx$ is no equivalence relation since it lacks ...
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1answer
49 views

Equivalence between Category of Covers and $\pi_1(X)$ Sets

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 38): In order to show the category equivalence claimed in Thm 2....
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17 views

Gluing hyperbolic convex polygon

How can I prove the existence of a regular convex $4n$-gon, $C$, with the angles $\pi /2n$ and how would I be able to show that $C$, is a gluing diagram for a hyperbolic surface after is has been ...
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39 views

Unit circle and cone

I know that in the following question, I would consider finitely many cases depending on where the points are located, but I would appreciate help on this: a) How would I define a unit circle in a ...
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29 views

Proof set of content zero has void interior

Given a set of content zero $A \subset R^{n}$, how can be proved that the interior of $A$ is void?
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20 views

Cone and vector field

I have a euclidean cone, $A$, with some cone point $p$. Assume that a vector field on $A - p$ is parallel if an isometry that carries an open set $A - p$ to $\mathbb R^2$ carries the vector field to ...
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1answer
12 views

Show that a set $K$ cofinal in a directed set $J$ is also a directed set

$J$ directed means that for all pairs $a,b$ in $J, \exists c$ s.t. $a,b<c$ and $K$ is cofinal in $J$ so for each $j \in J$, $\exists d \in K$ such that $j<d$. I am struggling with how to prove ...
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24 views

Connected sum of Torus and Klein bottle.

If we look at the polygons of these spaces and we cut out a piece and glue it together like so, Do we lose any information when we compute the fundamental group using CW complex method described in ...
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2answers
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What happens if $U\in \textit{P}(X)$ is always either closed or open

Let $X$ be a topological space s.t. $\forall U\in \textit{P}(X)$, $U$ is either open or closed (or both only in the cases $U=X, U=\varnothing$). What can we say about $X$? I found this example of ...
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1answer
32 views

Properties of Minkowski product of sets

We work in a normed vector space $\mathbb{X}$. Let $A$ a subset of unit sphere $\mathbb{S}_1=\{ x\in \mathbb{X} \mid \|x\|= 1 \}$. Decide if: $A$ is compact iff $[0,1]A=\{ tx | x\in A, t\in [0,1] \}...
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1answer
36 views

Prove that for each cardinal number $\alpha > 0$ exists a topological space $X$ so that $ w(X) = \alpha$.

Prove that for each cardinal number $\alpha > 0$ exists a topological space $X$ so that $ w(X) = \alpha$, where $ w(X)$ is the minimal cardinality of all bases of the topology on X. Can I just ...
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1answer
29 views

Find $\mathbb{R}_l \cap \mathbb{R}_K$

Find $\mathbb{R}_l \cap \mathbb{R}_K$, where $\mathbb{R}_l$ is the lower limit topology on $\mathbb{R}$ and $\mathbb{R}_K$ is the K-topology on $\mathbb{R}$. Is the intersection an empty set? I can't ...
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46 views

Real Analysis: If the restrictions on two closed sets are continuous, then the function from the union of the two closed sets is also continuous. [duplicate]

Suppose that $A,B \subseteq \mathbb{R}$ are closed and that $f: A \cup B \rightarrow \mathbb{R}$. Show that if the restrictions $f|_{A}: A \rightarrow \mathbb{R}$ and $f|_{B}: B \rightarrow \mathbb{R}...
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1answer
33 views

Understanding the meaning of immersion for Manifolds

Let $f: X \rightarrow Y$ be a smooth map of manifolds. $f$ is an immersion at $x \in X$ if $df_x: T_x(X) \rightarrow T_y(Y)$ is an injective map where $y = f(x).$ $T_x(X)$ is the tangent plane of $X$ ...
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1answer
31 views

Is the preimage of a simple root connected?

Consider a matrix $A(p) \in \mathbb{R}^{N \times N}$ that depends linearly on a set of real nonnegative parameters $p =[p_1,\dots,p_n], p_j \geq 0$. Let $A$ be nonnegative irreducible for any ...
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Point Set Topology (open sets, polynomials, hausdorff spaces)

I am trying to complete this Topological problem and I have completed it, I just would like some opinions on how to make my work better and if someone could check it for me as well. I would greatly ...
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1answer
25 views

Example of Non-Compact Closed Set contained in Open Set With Special Property

I am looking for an example of an open set $A$ in a metric space $X$ and a closed, non-compact subset $B$ of $A$ such that there is no $\delta > 0$ s.t. $\{x: \textrm{dist}(x,B) < \delta\} \...
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0answers
36 views

Meaning of Path Connectedness

Recently I've been reading through Chapter 9 of Artin's "Algebra". I'm having trouble understanding the proof of a proposition (Proposition 9.7.9, to be precise). It states: "Let $G$ be a path-...
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1answer
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Show that every locally compact Hausdorff space is completely regular

It would be very appreciated if someone could review my solution. Thanks! Problem: Show that every locally compact Hausdorff space is completely regular. Proof: Let X be a locally compact ...
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1answer
35 views

If $f\in C_0$ and $g$ is continuous, can we show that $f\circ g\in C_0$?

Let $E$ be a locally compact Hausdorff space and $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\}.$$ Let $f\in C_0(E)$, $E'$ be another ...
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1answer
26 views

Topology, compact

I have had some problems to solve the following question : Is there a continuous bijection of a compact Hausdorff space over the space of rational numbers? I think the answer should be no, it has cost ...
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31 views

Compactness of a subset of $L_1([0,1])$

Suppose that $X = L_1([0,1]):= \{f:[0,1]\mapsto [0,\infty): \int_0^1 |f(x)|~dx <\infty\}$. Equip $X$ with the standard metric $d(f,g) = \int_0^1 |f(x)-g(x)|~dx$. Now, define: $$C:= \{f\in X: f(x) \...
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1answer
36 views

Addition of continuous functions over topological spaces is continuous.

Just wanted to verify if the following proof works: Suppose $f:X\rightarrow \mathbb{R}$ $g:X\rightarrow \mathbb{R}$ are continuous. Want to show that: $h=f+g$ is continuous. We are going to prove ...
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2answers
34 views

A Homology calculation Question

Let $q: S^n\rightarrow S^n\vee S^n$ be the map we get quotienting the equator. What is $q_*$ on $n$-th Homology level? $q_*:\mathbf{Z}\rightarrow \mathbf{Z}\oplus\mathbf{Z}$ ?
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Hausdorff, locally connected and locally compact space reference.

I would like to find a reference to the following proposition: Let $X$ a Hausdorff, locally compact, locally connected, connected space and $K \subseteq X$ a compact subset. Then, there exists a ...
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33 views

Convexity properties of a cone

Let $M$ be a normed space and $A$ a subset (nonempty) of the unit sphere ($S_1$, which is the points of norm $1$). Define, for $\alpha >0$, a $A^{\alpha}=\{x\in S_1 | d\left( x,A\right) \leq\alpha\}...
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I want to prove that a topological space is a category with elements as objects and homotopy equivalence classes of paths as morphisms.

I want to prove that a topological space $(X,T)$ is a category. Here $X$ is a the underlying set and $T$ a topology on $X$. 1) Let the objects be the elements of $X$. 2) For each $x,y \in X$ the ...
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1answer
35 views

Showing isomorphism of two $C^*$ algebras

It seems that quite a standard trick of showing two $C^*$ algebras are as follows: Let $A$ be a $C^*$ algebra $B$ another $C^*$ algebra. $A' \subseteq A$ be a subalgebra that is closed under $*$. (...
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1answer
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A question about open equivalence relations

Definition. An equivalence relation $\sim$ on a topological space $X$ is said to be open if the projection map $\pi\colon X\to X/\sim$ is open. Let $U$ be an open set in $X$ and consider the set \...
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3answers
38 views

Countable open cover $\{U_i\}$ with each $U_i$ second countable [duplicate]

Let $X$ be a topological space that admits a countable open cover $\{U_i\}_i$ such that each $U_i$ is second countable in the subspace topology. Show that $X$ is second countable. My attempted proof ...
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1answer
46 views

Singletons in the $\sigma$-algebra generated by clopen sets of a Stone space

Let $A$ be a Boolean algebra and let $Ult(A)$ be its Stone space, that is, the set of all ultrafilters on $A$. It is well known that $C=\{\{u\in Ult(A)\!:a\in u\}\!:a\in A\}$ is an algebra of sets ...
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1answer
50 views

Is Hausdorff condition necessary to solve this problem?

Suppose that $X$ is a compact Hausdorff space and let $\mathcal{F}\subseteq\mathcal{P}(X)$ be a family of closed sets in $X$ whit the FIP property. Let $U\subseteq X$ be an open set such that $\bigcap\...
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1answer
50 views

Is $X$ path connected?

Let $X$ be union of lines $\{1/n\}_{n\in\mathbb{N}}\times\mathbb{R}$ and $\mathbb{R}\times\{1/n\}_{n\in\mathbb{N}}$, and the origin. Is $X$ path-connected? Attempt: My claim is that this is not path-...
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Second countability of Minkowski double cones.

In the 4-dimensional Minkowski spacetime, for a given point $x=(x^0,x^1,x^2,x^3)$, its timelike future or past set is defined as, $I^{\pm}(x)= \{y=(y^0,y^1,y^2,y^3) \in \mathbb{R}^4: \eta_{\mu \nu}(y−...
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1answer
35 views

If $H$ is a closed subgroup of a profinte group $G$, then $H$ is the inverse limit of the open subgroups of $G$ containing $H$.

Show that if $H$ is a closed subgroup of a profinte group $G$, then $H$ is the inverse limit of the open subgroups of $G$ containing $H$. I was able to show that $H$ is profinite. I use a ...
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2answers
42 views

Cauchy sequence in a discrete space

how to prove that any Cauchy sequence in a discrete space is stationary Let $(x_n)$ be a cauchy sequence then $$\forall \varepsilon>0, \exists n_0\in \mathbb{N},\forall p,q \geq n_0\Rightarrow \...