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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3
votes
2answers
56 views

Prove that $f_A (x) = d({\{x}\}, A)$, is continuous.

Prove that: Let $(X, d)$ be a metric space, and let $A$ be a subset of $X$. The function $f_A\colon X\rightarrow \mathbb{R}$, defined by $f_A (x) = d({\{x}\}, A)$, is continuous. Honestly, I ...
2
votes
0answers
39 views

Showing a set is closed or not in the space of real continuous function

Let $A=\{F\in C([0,1]\times[0,1]) : F(x,y)=f(x)+g(y)\}$ for some continuous functions $f,g:[0,1]\to\mathbb{R}$ Is $A$ closed in the space $C([0,1]\times[0,1])$ of real continuous functions with the ...
0
votes
2answers
32 views

Orientation of Edges on Graphs with Vertex Degree Constraints

Suppose I have a graph $ G = (V,E) $ such that each vertex $ v \in V $ has degree 4. Can I always choose an orientation of edges (ie. arrows drawn on edges) such that each vertex has two incoming ...
2
votes
1answer
25 views

Help with proof; Pre-image of a continuous function around a maximum is open.

Suppose a function $u$ is defined in an open and connected set $D$ and has maximum value $c$. Then if $u$ is not constant in D then the set $\{u(z) < c \mid z \in D\}$ is non-empty and open. This ...
0
votes
1answer
51 views

Open interval $(a,b)$ is open in $\mathbb R$ but not open in $\mathbb R^2$ and $[a,c) (a<c<b)$ open in $[a,b)$ but nor open or closed in $\mathbb R$

Open interval $(a,b)$ is open in $\mathbb R$ but not open in $\mathbb R^2$ and $[a,c) (a<c<b)$ is open in $[a,b)$ but nor open or closed in $\mathbb R$. $[c,b)$ closed in respect to $[a,b)$ but ...
1
vote
1answer
34 views

The boundary of the intersection of a decreasing sequence of convex sets

Let $\Omega_1 \supset \Omega_2 \supset \dots$ be a sequence of bounded, open and convex sets in $R^n$ with $\Omega :=\operatorname{int}(\overline{\bigcap \Omega_n})$ nonempty. It seems that $\partial ...
0
votes
1answer
33 views

Unit quaternion ball is compact and connected?

Let$$\mathbb{U} := \{x \in \mathbb{H} : |x| = 1\}.$$This is a group under multiplication. What is the easiest way to see that $\mathbb{U}$ is a compact and connected subset of $\mathbb{H}\cong ...
1
vote
4answers
65 views

Why isn't the initial topology always the trivial topology?

If I have a set $X$ and a function $f:X\rightarrow X$, then I think $f$ is continuous with the trivial topology, because no matter what the function is, $f(X)\subseteq X$. Thus for any point $f(x)$, ...
-2
votes
2answers
38 views

In $(X,d)$ metric space, an intersection of finite families of open sets is open. [duplicate]

Can anyone help prove this? I am allowed to used the fact that $X$ and $\emptyset$ are open and that a union of an arbitrary family of open sets is open. I tried to understand what it says in my ...
3
votes
3answers
69 views

If $(X,d)$ is a metric space, why are $X$ and $\emptyset$ open sets?

My definition of an open set $A$ in a metric space $X$ is that $\forall x\in A\exists y>0: B(x,y) \subseteq X$. How does this theoretically coincide with $X$ and $\emptyset$ being open, looking ...
4
votes
2answers
101 views

Showing a set is a compact subset of $\mathbb{R}$

Question: Let $$A=\{ x \in \mathbb{R}: x(x^{3}-3x-1)\leq15 \}.$$ Show that A is a compact subset of $\mathbb{R}$. I am just wondering how to approach this problem. Should I try showing it ...
2
votes
1answer
33 views

Can we always find $q\in\overline{\{p\}}$ with $p\notin \overline{\{q\}}$

Suppose $X$ is a quasiprojective scheme. Let $p\in X$ be a non-closed point, i.e. $\overline{\{p\}}\neq \{p\}$. Can we always find $q\in \overline{\{p\}}$ with $p\notin\overline{\{q\}}$? Of course, ...
0
votes
0answers
75 views

Homeomorphisms in $\mathbb{R}^2$

Consider $(0,0)$ in $\mathbb{R}^2$. Is there a neighborhood $U$ of $(0,0)$ open in the upper-right quarter plane but not in $\mathbb{R}^2$ (I mean such that if $(x,y)\in U$ then $x,y\ge0$) which is ...
-1
votes
1answer
37 views

Existence of exhaustion by compact sets

I am wondering when it is known that a set $A$ in topological space $X$ can be exhausted by compact sets, that is there exists increasing sequence of compact sets covering $A$. I guess this should ...
5
votes
1answer
55 views

Example of a surjective local homeomorphism that is not a covering? [duplicate]

Let $X$ and $Y$ be connected, locally path connected, and Hausdorff topological spaces. Can someone give me an example of a surjective local homeomorphism that is not a covering? I don't think this ...
0
votes
0answers
42 views

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 ...
1
vote
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, ...
0
votes
1answer
27 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- ...
1
vote
0answers
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 ...
3
votes
2answers
54 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 ...
2
votes
2answers
39 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 ...
2
votes
1answer
74 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 ...
3
votes
2answers
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 ...
1
vote
1answer
48 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 ...
0
votes
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 ...
2
votes
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?
8
votes
1answer
53 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 ...
1
vote
3answers
126 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, ...
2
votes
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; ...
0
votes
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 ...
0
votes
2answers
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 ...
0
votes
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 ...
2
votes
0answers
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 ...
7
votes
5answers
406 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 ...
2
votes
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 ...
3
votes
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 ...
22
votes
2answers
253 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 ...
3
votes
2answers
60 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 ...
1
vote
1answer
37 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 ...
0
votes
0answers
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 ...
3
votes
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 ...
4
votes
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 ...
-3
votes
1answer
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.
0
votes
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 $X$. ...
3
votes
2answers
69 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 ...
3
votes
0answers
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
2answers
61 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 ...
4
votes
3answers
151 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 ...