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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38 views

Whether $U\setminus C$ is open if $C$ is closed and $U$ is open

If $C$ is a closed set and $U$ is an open set, then is $U\setminus C$ open in an arbitrary metric space? I don't think this holds in the discrete metric space.
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1answer
44 views

How can I show that the image of this sequence is compact?

Let $(X,d)$ be a metric space, and let $(x_n)_{n=1}^\infty$ be a sequence in $X$ with limit $x_0$. Show that the subset $\{x_0, x_1, x_2, \dots\}$ of $X$ is compact. This is a book problem from A ...
2
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1answer
129 views

Unique smallest and largest topology ideas

What does it mean to be a smallest topology of a set $X$. I would guess that it would be a topology of $X$ which has least number of elements and similarly for largest topology it would have to be ...
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1answer
52 views

Isometries on the Banch Space M([0,1]) of regular Borel Measures

I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One ...
3
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1answer
93 views

Need a unique convergence (UC) space's Alexandrov extension be a UC space?

Background Say a topological space $X$ is a unique convergence (UC) space iff every sequence of points of $X$ converges to at most one point of $X$; a unique convergent clustering (UCC) space iff ...
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1answer
138 views

Long line is connected and compact

How to prove that the long line is connected and compact. I was trying to prove connectedness using contradiction but couldn't.
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1answer
246 views

$\sigma$-algebras and product topology

What can be said about $\sigma(T_1 \otimes T_2)$ and $\sigma(T_1) \otimes \sigma(T_2)$, when $T_i$ are topologies that aren't necessary second countable, and $\otimes$ denotes, at the left, the ...
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2answers
102 views

Topology Basis and open set

Let $X$ be a topology space and $A$ be a subset of $X$. For each $x\in A$, there is an open set $U$ containing $x$ such that $U$ is a subset of $A$. Show that $A$ is open in $X$. Solution: Let ...
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2answers
63 views

Basis in Topology confusion

I came across a lemma in some online notes where it says that, "Given a collection of elements of a basis ($\mathcal{B}$), they are also elements of Topology ($\mathcal{T}$) on $X$. How do we know ...
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72 views

Property of a CW complex

I need a little bit of help on this exercise: Let $X$ be a $CW$ complex. Then $X$ is connected if and only if $X$ is path connected
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1answer
269 views

What is the homotopy type of the affine space in the Zariski topology..?

I'm asking this question out of curiosity, as I was unable to come to a conclusion. Consider the affine space over an algebraically closed field, for concreteness we can work with $\mathbb{C}^{n}$. ...
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1answer
46 views

How to distinguish a connected set or a disconnected set?

I have some problems with this question? How to distinguish a connected set or a disconnected set ? Let $A=\left\{(x,y):0<x\le 1,y=\sin\frac1x\right\}$, $B=\{(x,y):y=0,-1\le x\le 0\}$, and let ...
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1answer
49 views

topological sequential space $(X,\tau)$

Suppose in topological space $(X,\tau)$ every countably compact is closed.Let $(X,\tau)$ be sequential space. (1): If every infinite subset $A \subseteq X$ is closed, will $A$ be discreet in $X$? ...
4
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1answer
73 views

Closed and bounded does not mean compact in general

Suppose in metric space $\Bbb Q$ I consider the subset $\{x\in\Bbb Q | 2<x^2<3\}$. I want to find a open cover for this set which does not have any finite subcover. How to do it ? Hint is ...
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0answers
27 views

Let $(X,\tau)$ be a $T_B$-space…

A topological space is called $T_B$ if every compact subset is closed. (I):$Let (X,\tau)$ be a $T_B$-space which is not countably compact, $\{x_n :n \in \omega\}$ a set without accumulation points, ...
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2answers
34 views

Question on Compactness

Let the metric space be the real numbers with the usual distance formula. Let $E$ be an open interval from $1/8$ to $2$. Then $E$ would be compact if every open cover of $E$ has a finite cover. I know ...
2
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1answer
507 views

Show that the boundary of a set equals the boundary of its complement

$\newcommand{\bdy}{\operatorname{bdy}}$ I'm trying to show that $\bdy(A) = \bdy(A^c)$. I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c ...
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1answer
182 views

Limit points of $\left\{\frac{1}{n}+\frac{1}{m}:m,n\in{\Bbb N}\right\}$?

Let $$S=\left\{\frac{1}{n}+\frac{1}{m}:m,n\in{\Bbb N}\right\}$$ and $S'$ be the set of limit points of $S$. All the results I've found on Google or Math.SE only give the following $$ ...
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3answers
225 views

What characterizes topological spaces where every open set is closed?

Motivated by the valuation topology on a discrete valuation ring, which has the above property, I want to know if there's some (subjectively, probably) nicer criterion for when a space has every open ...
0
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1answer
120 views

hardy-littlewood maximal function

I am following Stein's real analysis book and he defines the maximal function of an integrable function $f$ by $\displaystyle f^*(x) = \text{sup}_{x\in B} \frac{1}{m(B)}\int_B|f(y)|dy$ where the sup ...
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0answers
45 views

A topological space is called $T_B$ if…

A topological space is called $T_B$ if every compact subset is closed. According to therem ( I, II , III), how does the below theorem proof?? Let $(X,\tau)$ be a $T_B$-space which is not ...
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0answers
15 views

Let $(X,\tau)$ be a $T_B$-space which is not countably compact…

Let $(X,\tau)$ be a $T_B$-space which is not countably compact, $\{x_n :n \in \omega \}$ a set without accumulation points, $\mathscr{F}$ a uniform ultrafilter defined over $\{x_n : 0 < n < ...
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2answers
66 views

topological space is called $k$- space

A topological space is called $T_B$ if every compact subset is closed. Let $X$ be $T_B$ and $X^* = X \cup \{\infty\}$ be one-point compatification of $X$. A topological space is called $k$- space ...
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1answer
49 views

Proving that half an isometry is a homeomorphism

Let $(K,d)$ be a compact metric space and $f:K\rightarrow K$ such that $$\forall x \in K, \forall y \in K, d(f(x),f(y)) \geq d(x,y)$$ Prove that $f$ is a homeomorphism. What I managed to prove is ...
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3answers
247 views

If $E$ is a connected space, then so is $\overline{E}$.

I have to prove that if $E$ is a connected space, then so is $\overline{E}$ (the closure of $E$) a connected space. I tried to prove the contrapositive. So suppose that $\overline{E}$ is not connected ...
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1answer
37 views

topological K - space

A topological space is called $T_B$ if every compact subset is closed. Let $X$ be $T_B$ and $X^* = X \cup \{\infty\}$ be one-point compatification of $X$. A topological space is called $K$- space ...
0
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1answer
53 views

one-point compatification of $X$

A topological space is called $T_B$ if every compact subset is closed. Let $X$ be countable and $T_B$ and $X^* = X \cup \{\infty\}$ be one-point compatification of $X$. We want to show that when $X$ ...
0
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1answer
27 views

Topologically Mixing

Suppose you have a finite interval that maps to itself, is it possible for that map to be expanding? The problem for me is that for the map to be expanding, a subset must be constantly growing, but if ...
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2answers
184 views

Find an example about the connectedness.

Let $I=[0,1]$ and $I\times I$ be a subspace of $\Bbb R^2$. Find an example satisfying the following conditions: $A$ and $B$ are subsets of $I\times I$. $A$ and $B$ are connected. $(0,0), ...
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0answers
130 views

Subsets of R2 that are convex, closed, and have non-empty interiors?

Can someone give me some guidance with this problem? Thanks. Suppose that $A, B \subset \mathbb{R}$ are convex, closed, and have non-empty interiors. Prove that $A, B$ are the closure of their ...
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1answer
43 views

Is this a sufficient proof

Problem: We want to show that the open ball B centered at zero with radius less than one has a boundary where x^2+y^2=1. Proof? The closure of a set A is the union of A and its limit points. Set ...
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0answers
51 views

Connected sum of two surfaces is a surface?

IS the connected sum of two surfaces a surface? Im having hard time trying to see this. Can someone kindly help me? thanks.
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1answer
69 views

Why does there exist a unique quotient topology that makes a given surjective map a quotient map?

I'm reading Munkres on the quotient topology, pg. 138 It says that: If X is a topological space and A is a set and if $p: X \to A$ is a surjective map , then there exists exactly one topology on A ...
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1answer
211 views

Help in proof of Compactness implies limit point compactness.

Munkres' Topology says Theorem 28.1. Compactness implies limit point compactness, but not conversely. Proof. Let $X$ be a compact space. Given a subset $A$ of $X$, we wish to prove that if ...
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3answers
383 views

Showing continuity of partially defined map

There is a theorem in Note on Cofibrations by Arne Strøm. It says Let $A$ be a closed subspace of a topological space $X$. Then $(X,A)$ has the HEP if and only if there are (i) a neighborhood ...
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1answer
129 views

Intersection of int(cl) of open sets

$\newcommand{\cl}{\operatorname{cl}}$ $\newcommand{\i}{\operatorname{int}}$ Prove that if $A$ and $B$ are open, then $\i(\cl(A\cap B))=\i(\cl A) \cap \i(\cl B)$. One way implication is easy as we ...
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0answers
28 views

The homotopy between two monomorphisms

$X$ is a compact Hausdorff space and $E,F$ are two complex vector bundles on $X$. If $f$ and $g$ are two homotopic monomorphisms from $E$ to $F$, then can we find a homotopy $f_t$ such that $f_0=f, ...
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1answer
231 views

clearification of limit point compact

I'm having trouble with an example from Munkres dealing with limit point compactness. The example is as follows: Let $Y$ consist of two points; give $Y$ the topology consisting of $Y$ and the empty ...
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1answer
88 views

Does every map $\mathbb{R}P^n\rightarrow\mathbb{R}P^n$ lift to a pair of maps $S^n\rightarrow S^n$?

Question: Given a continuous map $f:\mathbb{R}P^n\rightarrow\mathbb{R}P^n$, is there automatically a continuous map $g:S^n\rightarrow S^n$ such that $f,g$ commute with the covering map ...
11
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1answer
143 views

Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that $$\forall a \in G , \forall x \in G ...
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1answer
64 views

How can we ensure that a space is a subset of locally convex topological space?

I am looking for fast ways to ensure that a given set is a subset of topologically locally convex space. I have already read the posts post1:seminorms-in-locally-convex-spaces, ...
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0answers
83 views

Products of homeomorphisms

I was wondering if there is a theorem like "If $f_i:X_i\to Y_i$ are homeomorphisms then $\prod_i f_i : \prod_i X_i \to \prod_i Y_i$ is a homeomorphism" for $I$ finite. What about $I = \mathbb N$? ...
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0answers
120 views

Is this a good proof?

I want to prove: If $K$ is compact $T_2$ then the extreme points of the unit ball of $C(K)$ are precisely the functions $f\in C(K)$ such that $|f(k)|=1$ for all $k\in K$. Here is my proof: Can someone ...
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3answers
59 views

relation between metric and topology

I have two questions regarding the relationship between metrics and the topology that they generate : First , if the metric changes then, is it necessary that the topology would also change ? ...
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1answer
493 views

Boundary of union of two sets equals the union of their boundaries [duplicate]

Plesae give some hint to solve the following problem: If $A$ and $B$ are two subsets of a topological space $X$ such that $\overline{A}\cap \overline{B}=\emptyset$, then $\partial(A\cup B)=\partial ...
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1answer
46 views

closure of a sequence of pairwise disjoint closed sets

Let $X_{i}$ be a sequence of pairwise disjoint closed homeomorphic copies of $T_{2}$ space . If $\bigcup _{i =1}^{\infty} X_{i}$ is closed, then what is the closure of $\bigcup _{i =2}^{\infty} ...
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3answers
167 views

A little confusion about compactness and connectedness

This question may be a bit simple or even naive for some people but it indeed confuses me for a long time. Thank you all if you provide any explanation. I know concepts: compactness means any open ...
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2answers
203 views

X is compact and Y is Hausdorff and connected prove a function is surjective

I need help proving the following. A function $f:X\to Y$ is an open map if whenever $U$ is an open subset of $X$, then $f(U)$ is an open subset of $Y$. Let $X$ and $Y$ be topological spaces. prove ...
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4answers
194 views

Algebraic varieties in $\mathbb{C}^n$ cannot have interior points

I know that the zero-set of a non-zero polynomial in $\mathbb{C}[x_1,...,x_n]$ can not have interior points, but I'm trying to find a proof that doesn't require a knowledge of complex analysis like ...
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1answer
78 views

Prove that the Moore plane is regular?

I am trying to prove that the topology on the upper half of the plane generated by the basis given by open balls $B((x,y), r)$ where $r \le y$ and $B((x,y), r) \cup \{ (x,0) \} $ where $r=y$ is ...