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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Showing that the center of a Möbius strip is homeomorphic to a circle?

Consider the Möbius strip as the quotient space obtained from $[0,1]\times[0,1]$ when identifying $(0,t)$ with $(1,1-t)$ for $0\leq t\leq1$. Now consider its center, that is, the image of the set ...
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9 views

quick question about a definition of homeomorphism set classes

Take $S$ a surface of general type. I want to define $Q$ the set of homeomorphism classes determined by the surface $S$. How can i define $Q$? I think that $Q$ is determined by all surfaces $S^{'}$ ...
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18 views

Difference between Euler characteristics of a Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g$. Let $U$ be the complement of $r$ points in $X$. The Euler characteristic of $X$ = $2-2g$. That I understand. But I'm confused about the ...
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125 views

Existence of a continuous function which does not achieve a maximum.

Suppose $X$ is a non-compact metric space. Show that there exists a continuous function $f: X \rightarrow \mathbb{R}$ such that $f$ does not achieve a maximum. I proved this assertion as follows: ...
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1answer
38 views

Zero-dimensional but not Hausdorff

Let's call a space zero-dimensional if it has a basis of clopen sets, and is $T_0$. Is there a zero-dimensional space that is not Hausdorff?
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39 views

$C$ is an uncountable set. Show that $B(x,\epsilon)\cap C$ is uncountable.

Suppose $X$ is a separable metric space, and $C$ is an uncountable subset of $X$. Prove that there is a point $x \in C$ such that for each $\epsilon>0$, $B(x;\epsilon)\cap C$ is uncountable. I ...
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1answer
24 views

Continuous function in order topology

I have to prove that for every $ \alpha$ in $[0, \Omega)$, there exists a continuous function $f$ from $[0, \Omega)$ to $\mathbb R$ such that the pre-image of $0$ is $\{α\}$. I really have no idea ...
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1answer
59 views

Some topologies are more equal than others

On MathOverflow 5 years ago, I answered a question about Awfully sophisticated proofs for simple facts. I answered Fürstenberg's topological proof of the infinitude of primes. While the answer ...
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1answer
9 views

Lower semicontinuous non-negative function on a locally compact Hausdroff space with a countable base

An extended real number is an element of $\mathbb R \cup \{-\infty, +\infty\}$. Let $X$ be a locally compact Hausdorff space with a countable base. An extended real valued function $f$ on $X$ is ...
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1answer
20 views

What can I say about $f:G \to Y$?

If $f:X \to Y$ be inyective where $X$ is nonempty perfect Polish space, $Y$ second countable and exist $G\subseteq X$ such that $G=\bigcap_{n \in \omega}D_n$ where $D_n$ is dense open sets and ...
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28 views

Pushout in $\mathsf{Set}$ where one of the maps is injective

From I.M. James' book General Topology and Homotopy Theory: Suppose we have a cotriad $$X \xleftarrow{\xi}W \xrightarrow{\eta} Y.$$ ... we might expect the pushout of the cotriad to be a ...
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1answer
38 views

Show that $\bar A = A \cup [(0,0), (0,1)]$

In $(\mathbb R^2, ||\cdot||_{\infty})$, let: $A_0 = ]0,1] \times \{0\}$ $A_n = [(\frac{1}{n}, 0), (\frac{1}{n}, 1)]$ for each $n \ge 1$. $A = \cup_{n=0}^{\infty} A_n$ It is required to prove that: ...
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1answer
45 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
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1answer
28 views

Hahn-Banach Thm for Normed Space [on hold]

Let $X$ be a normed Space. For $x \in X$ define $J(x)=\{f \in X^ * : f(x)=\|x\| , \|f\|=\|x\|\}$. Prove that $J(x)$ is not the empty set.
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25 views

Lie bracket question

I am wondering if this is correct. Suppose $X$ and $Y$ are two smooth vector fields which vanish at $p$: $X(p) = Y(p) = 0$. Also assume that $[X, Y](p) = 0$. Is it true that the derivative of the ...
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60 views

Am I a toroid or not?

I heard that topologically (whatever that is, me not good at math), a donut and a mug are the same thing. Both have a hole going all the way through them. I found this animation transforming one into ...
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1answer
23 views

Counter Example to Tietze Extension Property for Arbitrary Topological Space

Above is my question. My only issue is the final bit! For statements $1.$ and $2.$, the answer is true, since in both cases $Y$ is normal and we know that both metric and compact, Hausdorff spaces ...
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1answer
6 views

Reference request: Topological space of polygonal chains and its properties

I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains: image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License A polygonal chain can be ...
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1answer
6 views

Do the elements of a sequence converging to a point in the intrinsic core of a convex cone belong to the intrinsic core of the set eventually?

Let $X$ be a general Banach space and let $C\subset X$ be a convex cone. Consider a sequence $x_n$ in the affine hull of $C$ such that $x_n\to x$ for some $x\in icr(C)$, where $icr(C)$ denotes the ...
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40 views

Prove that topology over rationals is different [on hold]

Let basis of a topology be $\mathcal{B}=\{[a,b) \mid a,b\in\mathbb{Q}\}$. Then prove that the topology induced by $\mathcal{B}$ is different from the lower limit topology over $\mathbb{R}$.
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1answer
28 views

Is there an infimum of distance metric in Compact subspaces??

Here is my exercise that I'm struggling with for 2 days. Since I'm a beginner for topology It is so hard for me to solve. Plz help me...
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1answer
25 views

Which one is finer: standard topology or upper limit topology?

Out of these two which one is finer over $\mathbb{R}$? Standard topology Upper limit topology
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1answer
24 views

Problem about $\sin(1/x)$ in topology. (open and closed functions)

Let $f:(0,\infty)\to [-1,1]$ defined by $f(x)=\sin(1/x)$. Show that $f$ is continuous but neither open nor closed, where $(0,\infty)$ and $[-1,1]$ are a subspace of $\mathbb{R}$ with usual ...
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29 views

A problem about general topology. (homeomorphims)

Let $f:X \to Y$ homeomorphims and $A \subseteq X$ have the property that $A \cap A^{'}=\emptyset$. Then $f(A)\cap (f(A))^{'}=\emptyset$. Idea: Contrapositive. Suposse $f(A)\cap (f(A))^{'} \neq ...
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24 views

Proving that $S^n$ (n-sphere) is locally connected.

Definition: A space X is said to be locally connected at $x\in X$ if for any open set $U$ containing $x$, there is an open connected subset of $U$ (say $W$) containing $x$.$$x\in W\subseteq U$$ A ...
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12 views

compact inverse is compact in canonical homomorphism

Let $G$ be locally compact Hausdorff group. Let $N$ be a closed normal subgroup of $G$. Let $f:G\to G/N$ be the canonical homomorphism. I want to show that for every compact subset $C$ of $G/N$, there ...
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1answer
27 views

A space is Hausdorff iff the diagonal is closed [duplicate]

Let X be a topological space. Prove that X is a T2-space if and only if $A = \{ (x,x) : x \in X \}$ is closed in $X \times X$. I've been able to prove first part of the result but not the converse. ...
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2answers
28 views

Totally separated, but the clopen sets do not form a basis

Is there a space $(X,\tau)$ that totally separated (that is for $x\neq y \in X$ there is a clopen set containing $x$, but not $y$), and the collection of clopen sets does not form a basis?
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39 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
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Prob. 7, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: What are the necesary and sufficient conditions on the $a_n$ and $b_n$?

Let $\mathbb{R}^\omega$ be the set of all sequences of real numbers with the uniform metric $\tilde{\rho}$ defined as $$ \tilde{\rho}(x,y) \ \colon= \ \sup \left\{ \ \min \left\{ \ \vert x_n - y_n ...
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1answer
41 views

Prob. 6, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: How is this set not open?

Let $\mathbb{R}^\omega$ denote the set of all sequences of real numbers, let $\tilde{\rho}$ denote the uniform metric on $\mathbb{R}^\omega$ defined as $$ \begin{align*} \tilde{\rho}(x,y) \colon= ...
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21 views

Product Spaces: Tube Lemma

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. My professor asked to prove the following Lemma. The Tube Lemma: Let $K$ be a compact ...
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1answer
24 views

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph

Let $X\subset\mathbb{R}^3$ be the union of the coordinate axies, I want to show that $\mathbb{R}^3-X$ is homotopy equivalent to a graph, and the question asks further "which graph" Let ...
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0answers
38 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
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2answers
44 views

Problem of General topology (Schaum)

Let $\tau=\{U \subseteq \mathbb{Z}_+ :$ if $n \in U \implies$ all divisor of $n $ is in $U \}$. Let $f: (\mathbb{Z}_+,\tau) \to (\mathbb{Z}_+,\tau)$. Prove: $f$ is continuous $\leftrightarrow$ if ...
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1answer
25 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
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0answers
38 views

Uncountability of $\mathbb{R}^I$ if $I$ is uncountable

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to ...
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1answer
13 views

Is the co-limit of a chain of normal subspaces necessarily normal?

Suppose $ X_0 \subset X_1 \subset X_2 \subset \dots$ is a chain of normal subspaces of $X$ such that $X= \cup_{i=1}^{\infty} X_i$. Assume that $X$ has the colimit topology w.r.t. these subspaces. Can ...
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23 views

Prove $X \setminus \operatorname{Cl}A = \operatorname{Int}(X \setminus A)$

Definitions ($X$ is a topological space): • $\operatorname{Cl}A$ is the intersection of all closed subsets of $X$ which contain $A$ as a subset. • $\operatorname{Int}A$ is the set of all ...
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1answer
27 views

Simple homotopy construction

I'm sure this isn't too difficult but i can't seem to do it if you have two loops $p_0 = e*g $ and $p_1 = g*e$ where $e$ is the trivial loop How would i construct an explicit homotopy between the ...
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1answer
27 views

Is the following a valid characterisation of complete metric spaces?

A metric space $(X, d)$ is called complete if and only if every Cauchy sequence converges. Now does the following hold: A metric space is complete if and only if every sequence $(x_i)_{i\in\mathbb ...
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1answer
18 views

Closure of $\{ ( e^t\cos t,e^t\sin t) : t \in \Bbb R \}$

Suppose $\alpha: \Bbb R\to \Bbb R^2$ given by $\alpha (t)=(e^t \cos t,e^t \sin t)$, $A=\alpha(t)$ is a smooth manifold. What is the closure of $A$? I know that the closure of the set is the set ...
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1answer
34 views

$\mathbb{Q}$ with topology from $\mathbb{R}$ is not locally compact, but all discrete spaces are

Wikipedia claims that all discrete topological spaces are locally compact but that $\mathbb{Q}$ isn't when endowed with the topology of $\mathbb{R}$. I don't know if I understand those examples ...
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18 views

Weak topology and the closed unit ball

I want to prove that there is no neighbourhood of $0$ in the closed unit ball. I can use pointwise and Banach-Alaoglu theorems to prove it.
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1answer
30 views

Tietze Extension Theorem ,,

If we have X a normal space, C a closed subspace of X, Y a completely regular space, and $f:C \rightarrow Y$ a continuous function. How do we show that f has a continuous extension $F: X \rightarrow ...
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1answer
32 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
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Subgroup $H \leq G$ acting on $G$ by translation is transitive?

In Elementary Topology. Textbook in Problems, by Viro, et al they state the following: Let $G$ be a topological group, $H \leq G$ a subgroup. Then $G$ is a homogeneous $H$-space under the ...
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1answer
30 views

if $V_1\cong U_1, V_2\cong U_2$, is $(V_1\cup V_2 \cong U_1\cup U_2)$? Pasting homeomorphisms

My question arises from the theory of covering spaces. assume $f:Y\to X$ is a covering map, or more generally a local homeomorphism. Assum $U_1,U_2\subset X$ are open sets such that $f|_{V_1}, ...
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32 views

Prove that this infinite sum involving metrics is also a metric

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho_i: X\times X\to \Bbb R^+$ with ...
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2answers
45 views

Prove that this is a metric space

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho: X\times X\to \Bbb R^+$ be a metric on ...