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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Ribbon Surfaces and Legendrian Graphs on Contact 3-manifolds.

Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon ...
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52 views

Limit points of infinite subsets of closed sets

Is the following statement true or false? If $F$ is an infinite subset of a closed set $E$, then $F$ has a limit point in $E$? The original one is: if $E$ is an infinite subset of a compact set ...
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65 views

Non-compactness of $\mathbb{R}$ with the cocountable topology

Is $(\mathbb{R},\tau_{co})$ compact where $\tau_{co}$ is the cocountable topology on $\mathbb{R}$? I have the answer of my teacher but I'd like to see another one so I can understand better how ...
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56 views

Infinite spaces in which all subsets are compact are not Hausdorff

Let $(X,\tau)$ be an infinite topological space with the property that every subspace is compact. Prove that $(X,\tau)$ is not a Hausdorff space. I start by supposing $X$ is Hausdorff. Then I can ...
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59 views

Uniqueness of continuous extension from $A$ to $\overline{A}$ for maps into a Hausdorff space

I want to prove the following. Let $A$ be a subset of $X$. Let $f:A \to Y$ be continuous. Let $Y$ be Hausdorff. Show that if $f$ can be extended to a continuous function $g:\overline{A}\to Y$, ...
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33 views

Find the point implied by intermediate value theorem

Consider a function $f(x)$ such that $f(0)=0$ and $$f'(x) = \frac{T-x}{T-f^{-1}(x)} + \frac{T-x}{S}$$ Then we can see that $f'(0)>1$ and $f'(T)=0$. Find $x$ such that $f'(x)=1$, in terms of the ...
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96 views

Convex interior topology

I have found a fascinating example of topology on a vector space $V$, but I cannot prove its interesting properties to myself. Let $\mathcal{B}$ be the family of all convex symmetric (i.e. $\forall ...
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1answer
56 views

A continuous bijection from a Hausdorff space to a non-compact space which is not a homeomorphism

Recall the following theorem: Let $X$ be a compact space and $Y$ a Hausdorff space. Suppose that $f:X \rightarrow Y$ is a continuous bijection. Then f is homeomorphism. Prove that the ...
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60 views

Showing that a space is normal and not locally compact

Let $E$ be the set of all ordered pairs $(m,n)$ of non-negative integers. Topologize $E$ as follows: For a point $(m,n)\neq (0,0)$, any set containing $(m,n)$ is a neighbourhood of $(m,n)$. A set ...
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78 views

The Cantor set is not strong measure zero

$A \subseteq \mathbb R$ is strong measure zero if given any sequence $( \epsilon_n )_{n \in \mathbb N}$ of positive reals there is a sequence $( I_n )_{n \in \mathbb N}$ of open intervals such that ...
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43 views

Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
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1answer
51 views

Compact opens in sober $T_1$ are closed?

I am trying to establish some basic facts about spectral spaces. In relation to this I am looking for a proof of, or a counter example to, the statement that compact open subsets of a sober $T_1$ ...
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1answer
48 views

Cover a sphere by two closed subsets not containing a closed self-antipodal connected subset?

Question (Fulton's Algebraic Topology, A First Course, Problem 4.40) Suppose the sphere $S^2=A\cup B$ where $A,B\subseteq S^2$ are two closed subsets of $S^2$. Is it true that either $A$ or $B$ must ...
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31 views

Is $(\mathbb{R},\tau_B)$ a separable space?

Is $(\mathbb{R},\tau_B)$ a separable space? $\tau_b$ is the topology generated by $$\mathcal{B} = \{ \ [a,b) \ \ : \ \ a,b\in\mathbb{R}, \ a<b\}$$ I guess it's not separable ...
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49 views

Noetherian toplogical space exercise

Let $X$ be a noetherian topological space. Prove the following statements: (a) If $F \subset X$ is closed, then there exist $n \in \mathbb N$ and irreducible closed subsets $F_1,\ldots,F_n \subset ...
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1answer
29 views

Equivalent conditions for a topological space to be Noetherian

Problem Show that the following statements are equivalent: (a) $X$ is a noetherian topological space (b) Every non-empty family of closed subsets of $X$ has a minimal element. (c) If $$U_1 \subset ...
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64 views

Openness of a subset in complex 2-plane

Let $U$ be a subset of $\mathbb{C}^{2}$ containing the origin $0$. Assume that for any curve $C$ (an affine variety of dimension 1, maybe singular) passing through $0$ we have $U \cap C$ is ...
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3answers
53 views

Should real functions be described as $ f: \mathbb{R} \rightarrow \mathbb{R} $ or $ f: \mathbb{R} \rightarrow \mathbb{R}^2 $?

I've been trying to teach myself topology, and I'm having a bit of trouble grasping the abstract concepts of the field. One question that's been poking at my understanding regarding topological ...
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1answer
79 views

Prove Two Topologies Equivalent

I was reading Lawson's Topology as review and stumbled across this: For $X \subset \mathbb{R}^n$, show that the usual topology on $X$ is the same as the subspace topology. Here the usual ...
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1answer
48 views

What's the 1-dimensional topology of a graph?

I'm reading through this paper here downloads.hindawi.com/journals/mpe/2013/815035.pdf where they say "Since a graph can be equipped with a topology to turn it into a a one-dimensional space, we can ...
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1answer
50 views

Accumulation point in topological space problem

Exercise If $(x_{\alpha})_{\alpha \in \Lambda}$ is a net, then $x$ is an accumulation point of the net if for every $A \in \mathcal F_x$, the set $\{\alpha \in \Lambda: x_{\alpha}\in A\}$ is cofinal ...
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107 views

Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
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1answer
32 views

What is the induced functor of covering spaces to covering groupoids?

I'm reading May's book, 'A Concise Course in Algebraic Topology' and I'm confused about what he means by the induced functor from a covering space. First, here are some helpful/relevant definitions. ...
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4answers
135 views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
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2answers
47 views

Construction of a small but fat set? [duplicate]

Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of $A$ minus its interior is positive ?
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49 views

Show that $X$ is Hausdorff if and only if the diagonal $\Delta = \{(x, x):x \in X\}$ is closed in $X \times X$

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify it or offer suggestions for improvement? Show that $X$ is Hausdorff if ...
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46 views

Find the closure of $\mathbb{R}^{\infty}$ in $\mathbb{R}^w$ under the box topology

Find the closure of $\mathbb{R}^{\infty}$ in $\mathbb{R}^{\omega}$ under the box topology. Note: $\mathbb{R^{\infty}}$ is the set of all sequences $(t_1,t_2,...)$ such that $t_i\not=0$ for only ...
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34 views

Proving no finite basis of the system of neighborhoods at $a$ in the real line exist.

I'm not sure how to prove it, the gist is: I need to find the "smallest" neighborhood in the basis, take a ball of half that radius and show "look, there is no member of the basis in this ball, thus ...
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69 views

Perfect Map $p:\ X\to Y$, $Y$ compact implies $X$ compact

I was assigned the following homework problem for a introductory course in topology: Let $p:\ X\to Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact for each $y\in Y$. ...
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23 views

Intuitive affirmation on convex sets

Let $D_1, D_2$ two open, bounded and convex domain in $R^n$. Suppose that $D_2 \supset \overline{D_1}$, and the boundaries of these sets are of class $C^1$. Fix $x \in \partial D_1$ and suppose that ...
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1answer
22 views

Equality in product topology spaces.

I have the following problem: Given $A\subset X$, $B\subset Y$ topological spaces then $$\partial (A\times B)=(\partial A \times \bar B) \cup (\bar A \times \partial B) $$ I have no clear ...
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26 views

intuition on the countable dense subset implying separability [duplicate]

Are there any good intuitions to understand why countable dense subset implies separability? Is the separability related to the opposite of connectedness? In Munkres, I am a bit confused after he ...
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73 views

Why are clopen sets a union of connected components?

The wikipedia page on clopen sets says "Any clopen set is a union of (possibly infinitely many) connected components." I thought any topological space is the union of its connected components? Why ...
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67 views

Finding the closure of a subset

I have that problem: We have $(\mathbb{R}^2,\tau)$, where $\tau$ is the standard topology. Find the closure of $$A = \{ (x,y)\in\mathbb{R}^2\ \ |\ \ x^2+y^2<1 \}$$ I know that the boundary of ...
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1answer
71 views

The nature of isomorphism between fundamental groups with different base points

New to algebraic topology. Munkres (Topology, 2 ed.) in the last paragraph on page 332 says that "If $X$ is path-connected, all the groups $\pi_1(X,x)$ are isomorphic, so it is tempting to try to ...
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1answer
99 views

The map $t\mapsto (\cos t,\sin t)$ is injective from $[0,2\pi)$ onto the circle, but its inverse is not continuous

Question: given $\phi:[0,2\pi[\mapsto\mathbb{R}^2$ a map defined by $\phi(t)=(\cos t,\sin t)$ then Shown that $\phi$ is injective into unitary circle $S^1=\{(x,y)\in\mathbb{R}^2\mid x^2+y^2=1\}$ ...
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1answer
47 views

If $G/H$ and $G$ are connected linear algebraic groups must $H$ also be connected?

Let $k$ be a perfect field (e.g of characteristic zero) and let $G$ and $H$ be linear algebraic groups over $k$, with $H$ a normal subgroup of $G$. If both $G$ and $G/H$ are connected, must $H$ ...
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1answer
42 views

Does $X \subseteq \mathbb N^{\mathbb N}$ non-countable and $F_{\sigma}$ imply that $X$ contains a perfect set?

I think that the claim below is true, but whenever I try to prove it, I find myself using the continuum hypothesis ($\aleph_1 = \mathfrak c$). My question: Can the following statment be proved ...
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3answers
85 views

non-Borel subset of uncountable Tychonoff space

Let $X$ be an uncountable Tychonoff space. Must there exist a non-Borel subset of $X$?
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39 views

Topological space of continuous function is not compact

I'm struggling with this question: Let $C[0,1]$ be set of continuous function of $[0,1]$. Define metric $d(f,g)=\int^1_0|f(x)-g(x)|dx$. Show that $C[0,1]$, with topology $\tau$ induced by $d$, is ...
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48 views

Boundedness of continuous functions on compact sets

Let $E$ and $F$ be two metric spaces. If $K$ is a compact subset of $E$ then a continuous function $f:K\to F$ is always bounded and reachs its maximum. What happens if we replace $K$ by a closed ...
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43 views

Show that for $(X,d)$ a metric space, $U= \{x \in X: d(x, C) \leq d(p, C)\}$ is a closed set

Let $(X,d)$ be a metric space, $C$ be a closed set in $X$. Define $$d(C, x) := \inf \{d(c, x): c\in C \}$$ for all $x \in X$. Fix a point $p \in X$. Show $U= \{x \in X: d(x, C)\leq d(p, C)\}$ is a ...
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39 views

Closure of subspace topology

Let $X$ be a topological space and $A\subseteq X$ a subset of $X$. Let $Y$ be a subspace of $X$ so that $A\subseteq Y$, and let $A_X$ and $A_Y$ be the closure of $A$ in $X$ and $Y$ respectively. I ...
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1answer
80 views

The spectrum of a commutative ring with unity and its “topology”

Let $\operatorname{Spec}(R)$ be the set of prime ideals in the commutative ring with unity $R$, and let $\mathfrak a$ be some ideal. Show that we get a topological space if we define the closed sets ...
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34 views

Nowhere dense set - coarser vs. finer topology

Let $X$ be a set and let $\tau_1\subseteq\tau_2$ be topologies on $X$. Suppose that $A\subseteq X$ is nowhere dense in $\left(X, \tau_2\right)$. I was wondering if it follows that $A$ is nowhere dense ...
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37 views

Closed and open sets in metric spaces

I'm going to be a freshman and I have just learnt topology recently. Here is my question: If a set has no limit point, then it is closed? For example, from which I read in Principles of mathematical ...
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1answer
37 views

Unique Limits in T1 Spaces

It's intuitive to me that limits in T2 (Hausdorff) spaces are unique: $x_n \rightarrow l$ if you can find an $N$ such that for $n > N$, $x_n \in O$ where $O$ is any open neighborhood of $l$ and ...
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1answer
86 views

Exercise from Morris's book

I'm beginning to study topology using the Munkres's book, and also the Morris's book Topology without Tears. From the last book, I try to resolve some items of the Exercise 1.1.9, Chapter 1, Pag. 28: ...
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27 views

Embedding a space in its cone

Let $X$ be a topological space, and $C(X)= (X \times [0 ,1])/(X \times {1} )$, define $f\colon X \to C(X)$ as $f(x)=[x,t]$ for some fixed $t$ s.t $\ 0\leq t <1$. I have to show this is a ...
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2answers
67 views

Is the constant map a continuous function?

I've been set a question in an assignment which reads: "Check whether the following functions are continuous or open. Check whether they are a homeomorphism. $\dots$ $b)$ the constant map $f:X ...