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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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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24 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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40 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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29 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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46 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
28 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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62 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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51 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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43 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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43 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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89 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
31 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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81 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
44 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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1answer
37 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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32 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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30 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
53 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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19 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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44 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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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 ...
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1answer
55 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
79 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
43 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
41 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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75 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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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 ...
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1answer
45 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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1answer
40 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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31 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
79 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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33 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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29 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
30 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
81 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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1answer
23 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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54 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 ...
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54 views

First Course in Topology: Countable sets.

Consider nonempty sets $X$ and $Y$, and a function $f:X\rightarrow Y$. Suppose the inverse image of $Y$ under $f$ is countable for each element of $Y$ and assume $Y$ is countable. Prove that $X$ is ...
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22 views

Open subset of the space of matrices

This question comes from the process of my learning about Grassmann manifolds. Suppose that $M(m,n)$ is the set of real $n \times m$ matrices, where $n>m$. Let $F(m,n)$ be a subset of $M(m,n)$ ...
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123 views

Homology and topological propeties

i have this theorem with it's proof but i don't understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where ...
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47 views

The number of connected components of $SL(2, \mathbb{R})$ which keep $x^2 - y^2$ invariant

Working on yet another past comprehensive exam question. Let $S$ be the set of real $2\times 2$ matrices with determinant $1$, keeping invariant the form $x^2 - y^2$. Regard $S$ as a subset of ...
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1answer
85 views

Unit circle cannot be well ordered?

This showed up while reviewing for an analysis qualifier. I thought any non empty set has a well ordering through the Axiom of Choice. Clearly I am not understanding how making the order be compatible ...
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54 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
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16 views

Family of functions depending continuously on a parameter space WRT the $L^1$ norm

The material I'm reading involves a family of functions induced by a parameter space homeomorphic to an open disk. It attempts to show that the functions depend continuously on this parameter with ...
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1answer
78 views

Connectedness and compactness of K-topology

Let $T_K$ be the K-topology on $\mathbb{R}$, this is, the topology generated by the collection of all open intervals $(a,b)$ and the sets of the form $(a,b)-K$, with $K=\{1/n, n \in \mathbb{Z}^+\}$. ...
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1answer
126 views

Compact subspace of a covering space

I've been working through Massey's A Basic Course in Algebraic Topology and I've gotten stuck on the following exercise (V.8.4): Let $X$ be a regular topological space, and $(\tilde{X}, p)$ a ...
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35 views

What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...