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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How to understand this sentence within Atiyah-Macdonald's textbook about commutative algebra

In page 102 of this textbook, authors mentioned that: Assume topological group $G$ has a fundamental system of neighborhoods consisting of subgroups as: $G= G_0 \supseteq G_1 \supseteq\cdots\supseteq ...
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1answer
47 views

Every quasi-compact scheme has a closed point

I know this question has been asked here before, but I have trouble understanding the following proof, taken from a Schwede's write-up. I have underlined the bit I don't understand. In particular, ...
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1answer
28 views

A circle function $f : S^1 \rightarrow S^1$ has degree $0 \iff f$ extends to a continuous function on the disk $D$. Explanation?

The following theorem is an extension theorem concerning circle maps with degree $0$: It is hard to me to understand it, though! Would someone please guide me about the following questions: 1- ...
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21 views

Is the Zariski topology equipped with Eisenstein's metric an analytic submanifold?

Using $M=(C(\mathbb{R}),T_z)$ with the norm $(x,y) \to \log(\partial_x+\partial_y)$, we can easily define a derivative using distributions. I was wondering: Does this make $M$ an analytic manifold ...
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56 views

Proving characterization of continuity with direct images of sets using nets.

We know that if $X,Y$ are topological spaces, then $f: X \to Y$ is continuous if and only if $f(\overline{E}) \subseteq \overline{f(E)}$, for all $E \subseteq X$. I started studying nets by myself ...
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2answers
40 views

Existence of an metric or a topology so that every subset is compact

Let $X$ be a infinite set. Is there a metric on $X$ such that every sub set of $X$ is compact? What about a topology on $X$? I think that if we can answer first question then we can answer the ...
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1answer
79 views

Question about the fundamental group of a connected, open subset of $\mathbb{R}^2$

Let $U \subset \mathbb{R}^2$ be open and connected. Suppose $f: I \to U$ is a loop with $a = f(0) = f(1)$ such that $f$ doesn't wind around any $p \in \mathbb{R}^2 \setminus U$. a) Is it true that ...
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38 views

Prob. 4, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: For $T_1$-spaces countable compactness is equivalent to limit-point-compactness.

Definition: A topological space $X$ is said to be countably compact if every countable open covering of $X$ has a finite subcollection that also covers $X$. Definition: A topological space ...
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49 views

The union of family of open disjoint non empty subsets is dense.

The statement I'm gonna write down here is a proposition I conjectured in order to prove a detail in Hartogs theorem; thus I won't write all the context, otherwise it would become the longest post ...
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1answer
48 views

Characterization of analytic functions

First, see this link on the alternative characterizations of analytic functions. I want to prove a version of 3) for complex-analytic functions. In particular: If $f$ is a complex-analytic function ...
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1answer
81 views

Where does the “CW” in CW-complex come from?

I've heard people say that the "CW" in CW-complex comes from the "CW" in JHC Whitehead, though nobody has ever given me a reference for this. Does anyone know where the "CW" in CW-complex comes from?
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55 views

A manifold such that its boundary is a deformation retract of the manifold itself.

If we have a compact orientable manifold $M$, we know that $\partial M$ is not a deformation retract of $M$. This follows from Poincaré Duality or Stokes Theorem. If we take away compactness, this is ...
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3answers
127 views

a good modern topology book

I want to study an advanced modern book on topology, but I couldn't find any. I've already studied the first chapters of Munkres' book, but it is not as advanced as books such as Engelking's topology, ...
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1answer
49 views

Graph theory : the adjacency matrix of an n-dimensional torus

Is there, in principle, an easy way to determine the adjacency matrix of an n-dimensional torus that's only connected to neighbours which it shares a corner/edge/face/volume/etc with (n>1 obviously; ...
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1answer
24 views

Compact set on functions space

Let $(D[0,T], X\times X)$ the set of cadlag functions from $[0,T]$ to $X\times X$. If I have a compact subset $K$ in $(D[0,T], X)$ and another compact subset $H$ in $(D[0,T], X)$, is $K\times H$ a ...
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26 views

A proposition of relative interior point

One proposition from Convex Optimization Algorithm p.473: $X$ is a nonempty convex subset of $\mathbb{R}^n$ $f:X \rightarrow \mathbb{R}$ is a concave ...
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1answer
28 views

Hatcher algebraic topology book prop. 2.29

I am studying algebraic topology from Hatcher book and i don't understand the first sentence of proof of proposition 2.29. on page 135 , Proposition 2.29. $ \mathbb{Z}_2$ is the only nontrivial ...
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47 views

Filters, nets and Galois correspondence

In the lecture, our prof. mentioned that the correspondence between nets and filters is a Galois correspondence without giving any more details about that. In algebra, the proof of the Galois ...
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5answers
407 views

Necessity of being Hausdorff in the definition of compactness?

According to R Engelking - General Topology: A topological space $X$ is called a compact space if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover, i.e., if for every ...
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0answers
35 views

How to speak on limit of sequence categorically? [duplicate]

I was thinking on ways to define limit of a sequence (over the reals, or over a metric space, or even better, over a general topological space) using the categorical limit (final or inicial object of ...
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2answers
35 views

Proving that $(X,\tau)$ is Hausdorff given a condition.

Let $(X,\tau)$ be a topological space such that for each $p \in X$ there is a continuous function $f:X \to \Bbb R$ verifying $f^{-1}(\{0\}) = \{p\}$. Then $(X,\tau)$ is Hausdorff. Welp, take $p,q ...
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254 views

Can you define arc length using a piece of string?

In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This ...
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2answers
61 views

Example of strict inclusion in continuity condition $f(\overline{A})\subseteq \overline{f(A)}$

One definition of continuity is the condition $$f(\overline{A})\subseteq \overline{f(A)},$$ for all $A\subseteq X$. To understand this condition better, I tried to find an example of a real-valued ...
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38 views

$(M,d)$ is complete iff all closed and countable subspace of $M$ is complete

A metric space $(M,d)$ is complete $\iff$ if every closed and countable subspace $F\subseteq M$ is complete. $\implies)$ For this implication I use a proposition that says: "If $X$ is a complete ...
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19 views

Topological Semi conjugacy between Henon map and Logistic Map

I am currently teaching myself dynamical systems and have come across a problem I am not quite able to figure out. More specifically, I am unable to find a conjugator function to establish a semi ...
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2answers
60 views

Pre-images and local homeomorphisms

I want to prove that if $f: M \to N$ is a local homeomorphism, then for all $y \in N$ we have $f^{-1}(\{y\}) \subset M$ closed and discrete. Here's the catch: this is from an exercise sheet from over ...
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1answer
44 views

Accumulation points in $T_1$ space.

Let $(X, \tau)$ be a $T_1$ space. If $S \subset X$, $S \neq \varnothing$ and $x \in S'$, then every open neighbourhood of $x$ contains infinitely many points of $S$. Once I saw a pretty easy proof by ...
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65 views

Prove that the set $\{(x,y): y/x = 1\}$ is closed as a subset of $\mathbb{R}^2$ [closed]

Prove that the set $\{(x,y): y/x = 1\}$ is closed as a subset of $\mathbb{R^2}$. Would prefer if I had step by step help.
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2answers
42 views

A sufficient condition for a set to be dense.

A subset $A$ of a topological space $(X,\tau)$ is said to be dense if $\overline A=X$. Prove that if for each open set $O\neq\varnothing$ we have $A\cap O\neq\varnothing$, then $A$ is dense in $X$. ...
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70 views

Perfect set without rational numbers

Sorry if this problem is repeated. Is there a nonempty perfect set in $\mathbb{R}^1$ which contains no rational number? Proof sketch: This set must be uncountable because any nonempty perfect set in ...
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58 views

Are there less trivial necessary and sufficient conditions?

Given an infinite set $X$ with the finite-complement topology, find a necessary and sufficient condition for a map $f:X\to X$ to be continuous. I came up with the condition that $\lvert ...
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1answer
57 views

On the equality of two sets (a doubt from Probability with Martingales).

Let $(S, \Sigma, \mu) $ be $([0,1], \mathcal{B}[0,1], Leb)$. Let $\epsilon(k)$ be a sequence of strictly positive numbers s.t. $\epsilon(k) \downarrow 0$. Let $V = Q \cap [0,1],$ the set of rationals ...
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1answer
31 views

What topologies are placed on the domain and range of the characteristic function?

under consideration is: $\mathbb{1}_{[0,1)}:\mathbb{R}\to\{0,1\}$ $$\mathbb{1}_{[0,1)}(x)= \begin{cases} 1,& 0\leq x<1\\ 0,& \text{otherwise} \end{cases}$$ My first question is that I don't ...
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62 views

Prove that General Linear Group is a topological subgroup.

First of all for $\mathbb{R}$ in my book it is written that: "$GL(n,\mathbb{R})$ is an open subset of euclidian $n^2$-space and that is the topology is given. Matrix multiplication is given by ...
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3answers
153 views

Accumulation points of sets

Determine all of the accumulation points of the following sets in $\mathbb{R}^1$ and decide whether the sets are open or closed or neither. I have two problems with the following problems first ...
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38 views

Prove that set is perfect

Let $E$ be the set of all $x\in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Is $E$ perfect? Proof: Here was proved that $E$ is closed set in $[0,1]$ (also in $\mathbb{R}^1$). ...
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41 views

Non-Principal Ultrafilters Confused!!

I've just started learning about filters and non-principal ultrafilters. I'm getting confused on the requirement: $U$ contains no finite subsets of $J$; where $U$ is the ultrafilter and $J$ is a set. ...
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3answers
81 views

Completeness of subset of metric space.

Let $\mathcal C[a,b]$ be the space of continuous function $f:[a,b]\to \Bbb R$ with supremum metric. Let $l,m$ be fixed real numbers. Prove that the subset of $\mathcal C[a,b]$ consisting of all ...
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1answer
39 views

Weak topology, strongly continuous function

I have a question about weak topology. Let $X,Y$ be a real Banach space. Let $\tau_{X,s},\tau_{X,w}$ be a strong(normed) topology on $X$, weak topology on $X$ respectively. $\tau_{Y,s},\tau_{Y,w}$ be ...
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1answer
31 views

Hausdorff space definition in terms of disconnected subsets

I've read the definition of a topological Hausdorff space (two distinct points have disjoints neighbourhoods) and of a disconnected space (it is the union of two disjoint nonempty open sets) Now I ...
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2answers
66 views

The set of [0,1] which decimal expansion consists of 4 and 7

I'm sorry if my question is repeated. Let $E$ be the set of all $x\in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Is $E$ contable? Is $E$ dense in $[0,1]$? Is $E$ compact? Is ...
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1answer
41 views

Example 3, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: How does $S_\Omega$ satisfy the sequence lemma?

Here's the sequence lemma: Let $X$ be a topological space, let $x \in X$, and let $A \subset X$. If there is a sequence of points of $A$ converging to $x$, then $x \in \overline{A}$. Conversely, ...
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43 views

Sequences in ${\Bbb C}^{\Bbb R}$ as an example of nets

The following is an excerpt from the Real Analysis by Folland, which is an introduction of nets: I can understand each statement about the example of $\Bbb{C}^{\Bbb R}$ though, I don't know why and ...
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1answer
26 views

$G$ is open in $(A,d)$ iff $G = A \cap U$ for some set $U$ that is open in $(M,d)$

I've been working through Real Analysis by Carothers and need some help understanding the proof of this Proposition. In the forward direction ($G$ open in $A \implies G=A \cap U$), the proof goes ...
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59 views

Order Properties on Open Sets

Considering the subset order on the open sets of a topological space, it seems natural to ask what kind of total orders exist as suborders of the subset order. One possibility is that each total order ...
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1answer
30 views

Is a real closed, bounded interval a locally compact Hausdorff space?

Does this hold? I've been confused by the statement of the Riesz-Markov-Kakutani representation theorem; that is, the formulation is as follows: Let $X$ be a locally compact Hausdorff space. For ...
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3answers
93 views

Show that the letters X and I (thought of as topological spaces) are not homeomorphic [duplicate]

I am reading Hajime Sato's: Algebraic Topology, an Intuitive Approach. His Sample Problem 1.3 is: Show that the topological spaces X and I are not homeomorphic. (Note that this requires a font where ...
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2answers
47 views

Definition of cartesian product in topology

In munkres' book, the general definition of cartesian product is the set of j tuples satisfying some conditions...... Which means that the cartesian product is a set of functions. How does this ...
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1answer
64 views

union of two connect sets in particular case

Let $(X,d)$ is a metric space and $A,B \subset X$ are connected and $A \cap B = \emptyset$ and $A^- \cap B \neq \emptyset$ ($A^-$ is closure of $A$) now prove or disprove that $A\cup B$ is ...
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44 views

What is the Euclidean topology on $\mathbb{R}^0$ like?

I am trying to prove that a topological space $(X,\mathscr{T})$ is a $0$-manifold if and only if it is a countable discrete space. In the process I have to show that there exist a homeomorphism from a ...