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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If $(X,\tau)^n$ is Hausdorf, is $(X,\tau)$ also Hausdorff?

If $(X,\tau)^n$ is Hausdorf, is $(X,\tau)$ also Hausdorff? I know that product of Hausdorff space is Hausdorff, but I want to know if this weaker converse of it is true. Thanks.
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1answer
39 views

What is a “closed subspace” of a topological space?

I was reading a proof online and it linked to a book by Munkres which says Every closed subspace of a compact space is compact. I dug out the book and searched the index for this term. ...
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2answers
33 views

Continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$

Can we have a continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$?
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2answers
26 views

Metrizability of quotient spaces of metric spaces

Suppose $X$ a metric space and $\sim$ an equivalence relation on $X$. Is the space $X/\mathord{\sim}$ metrizable? I think that the answer is no, but I could not arrive at a counterexample.
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1answer
33 views

Connectedness of both $Y \cup A$ and $Y \cup B$ where $A, B$ is a separation of $X -Y$

Let $Y\subset X$ be such that both $X$ and $Y$ are connected. Show that if $A$ and $B$ is a separation of $X-Y$, then $Y\cup A$ and $Y\cup B$ are connected. I found a proof for this problem in this ...
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5answers
443 views

Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?

I understand that when we talk about union of open sets, we introduce an index set which can be countable or uncountable. But could I not do the same for the intersection of open sets too?
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1answer
21 views

Path Connectedness argument for $SO(n, \mathbb{R})$

I am trying to prove path connectedness of $SO(n, \mathbb{R})$. I have seen several different proofs for the same. But I had a thought and wanted to know whether it would help in any way. I took two ...
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0answers
12 views

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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1answer
45 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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2answers
52 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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2answers
47 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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47 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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0answers
29 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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1answer
48 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
39 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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2answers
47 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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2answers
49 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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0answers
40 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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0answers
15 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
38 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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2answers
28 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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1answer
44 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
27 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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56 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
49 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
41 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
41 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
37 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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78 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
29 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
73 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
43 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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0answers
53 views

continuous function-constant [on hold]

I followed topology course and my lecturer give my a question like this. Topology space X is cofinite topology or cocountable topology, then $f: X \to \Bbb R $ is continuous iff $f$ is a constant ...
2
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1answer
35 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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30 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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2answers
28 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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1answer
36 views

Is this a connected set? [on hold]

Suppose that $O(n) = \{A: A$ is $n \times n$ matrix such that $A^TA = I\}$ Is $O(n)$ connected?
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1answer
52 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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0answers
18 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
20 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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24 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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2answers
42 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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0answers
37 views

*Continuous real function on a compact set is bounded and achieves greater value and less in that set.* [on hold]

Continuous real function on a compact set is bounded and achieves greater value and less in that set. I know how prove is done and how boundeness achieve greater value, please help prove someone as ...
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2answers
63 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 ...
2
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1answer
52 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
71 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
42 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
39 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
74 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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1answer
34 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 ...