0
votes
3answers
31 views

X :compact and continuous function $f(x)\neq x$

Let (X,d) be a compact metric space and $f:X\to X$ be a continuous function such that $f(x)\neq x,\: \forall x\in X$. Prove that there exists $\epsilon > 0$ such that $d(x, f(x))>\epsilon$, for ...
0
votes
2answers
23 views

Is sequence convergent in subspace of compact metric space?

Problem is as follow. Let X be a compact metric space and A be a closed subset of X. Prove that every sequence in A has a convergent (note: convergent in A) subsequence. It is from my note. My ...
0
votes
0answers
27 views

Distance from a point x to a Set A [on hold]

Let $A\subset \mathbb{R}^n$ be a non-empty open set and a point $x\in \mathbb{R}^n$. Define $\rho(x,A) = \inf\{\|x-y\|:y\in A\}$. (a) Show that $N(A,\varepsilon)=\{x\in ...
0
votes
0answers
30 views

fundamental group of $\mathbb{C^*}/\{e,a\}$

I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$. ...
1
vote
3answers
99 views

Topologist's Comb

I've already posted another question because I've got an assignment to finish (still got half a day left :/) and had to realise that I'm a bit lost. Anyway, the question has to do with the following ...
1
vote
3answers
61 views

Are curves closed in $\mathbb{R}\times \mathbb{R}$ with the standard topology?

Given the graph of the curve $y=\frac{1}{x}$, can we determine if the curve is closed or open in $\mathbb{R}^{2}$ with the standard topology?
1
vote
2answers
66 views

Fundamental Group of Punctured Plane

What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$? Clearly this is homeomorphic to the half cylinder , which is ...
0
votes
1answer
42 views

Basic topology questions with cantor's set

I have 3 questions in toplogy, one of which I managed to solve (but would appreciate input regardless) and 2 which are more difficult. I'd like a push in the right direction. Define $K$ as ternary ...
0
votes
1answer
35 views

Question on Quotient spaces [closed]

(i) Show that the quotient space $(S^{2} \times [0,1])/(S^{2} \times \{0\})$ is homeomorphic to the 3-disc $D^{3}=\{(x,y,z)\in \mathbb R^{3} \mid x^{2}+y^{2}+z^{2}\leq 1\}$ (ii) Let ...
1
vote
0answers
35 views

Product Topology & Homeomorphic [closed]

(i) Describe the topological space $S^{0}$ x $S^{0}$ Can I just say $S^{0}$x$S^{0}$={(-1,1)x(-1,1)}? I know the definition of a product topology is saying that XxY has a basis consisting of all ...
3
votes
1answer
63 views

Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
2
votes
1answer
37 views

Using Cantor's intersection theorem

Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$. (Hint: Set $F_1 = f(X), F_{n+1} = ...
0
votes
0answers
20 views

What is the classification theorem of simple Lie groups?

I've seen this thrown around a bit, but I can't find what the theorem actually states? Can anyone help?
1
vote
1answer
22 views

Showing a simple Lie group is connected and compact.

I'm working on a presentation on simple Lie groups and would like to show by example that the simple Lie groups are connected, but I'm not really sure how to do this. I'd also like to show that one of ...
2
votes
3answers
100 views

Prove a union of a finite number of closed sets is closed - methodology w.r.t. grading

I have to grade some homework in solving the above question (in the title) but I have a dilemma. QUESTION: Given de Morgan's law and the fact that the intersection of a finite number of open sets ...
1
vote
3answers
60 views

topology homework question

so I got this question for homework: let $x$ be a topological space and let $A \subset C$. one sets $\alpha(A) = \mathrm{Int}(\bar{A})$, and $\beta(A) = \overline{\mathrm{Int}(A)}$. Prove if $A$ is ...
2
votes
0answers
44 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
1
vote
1answer
27 views

Misunderstanding of an exercise on topology?

Let $\mathbb{R}$ be the set of reals with a topology $\mathcal{T}$ such, that each $x\in\mathbb{R}$ has a base of open regions ...
1
vote
3answers
36 views

Let $(M, d)$ be a metric space and $A\subset M$. Show that $U\subset A$ is open

The entire question is as follows: Let $(M,d)$ be a metric space and $A\subset M$. Consider the metric space $(A,d)$. Show that a set $U\subset A$ is open in $(A,d)$ if and only if there exists an ...
-1
votes
1answer
27 views

proving the equivalence of to metrics

Any hints as to how I can prove that $(\mathbb{R}^n,d_\infty)$ and $(\mathbb{R}^n,d_T)$ are topologically equivalent. Where $$d_\infty = \sup\{|x_i-y_i|, |x_2-y_2|,..,|x_i-y_i|\} $$ and $$d_T = ...
0
votes
1answer
41 views

Proving a basis exists

How would I show that there exists some set of open balls with rational radius and rational centre such that they are a subset of the reals.That is, $\exists p,q\in \mathbb{Q} $ and $ r,x \in ...
0
votes
2answers
40 views

Proving homotopy of paths

Let $f$ be a path in $X$ and $h:[0,1] \mapsto [0,1]$ a continuous mapping with $h(0)=0$ and $h(1)=1$. How can I prove that $f$ and $fh$ are homotopic relative to the endpoints?
1
vote
2answers
42 views

Dense Sets in R

I was told that $\mathbb{Q}$ is dense in $\mathbb{R}$. If you add in the limit points of $\mathbb{Q}$ for the closure of $\mathbb{Q}$, then that is all of $\mathbb{R}$. I was also told that ...
1
vote
1answer
21 views

How are linked rings homeomorphic to seperated links?

I'm currently reading "Geometry, Topology and Physics" by Mikio Nakahara. In his book there the following exercise: Show that two figure in figure 2.109(b) [see below] are homeomorphic to each ...
1
vote
1answer
24 views

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and vice versa

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and ...
0
votes
1answer
69 views

Separability of a normed space

The following is an Exercise 12, page 75 of conway's Functional Analysis. Let $\oplus_\infty X_i = \{x\in \sqcap X_i: ||x||=\sup||x(i)||<\infty\} $ where each $X_i$ is a normed space for $i\in I$. ...
0
votes
3answers
73 views

Closed subset of compact set is compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ...
2
votes
2answers
93 views

Problem about metric space

Let $X$ be the set of sequences of zeros and ones. For $x=(x_1, x_2, x_3, \dotsc)$ and $y=(y_1, y_2, y_3, \dotsc)$ in $X$, define $$d(x,y) = \sum_{j=1}^\infty \frac{ |x_j - y_j| } {2^j}.$$ (a) Prove ...
1
vote
2answers
42 views

Prove that $X-\bar S = int(X-S)$

I'm sorry for not having any sketch. I'm not be accustomed to topology. $\bar S$ : closure of S. $S\subset X^{metric}$. Prove that $X-\bar S = int(X-S)$ I add my proof. I think I have some ...
0
votes
4answers
49 views

topology about isolated and limit point

As a first class in topology, it is hard to prove. Can you help me? $S\subset X^{metric}$. Let $S_1$ be the set of limit points of S. Let $S_2$ be the set of isolated points of S. Show that $\bar S = ...
0
votes
1answer
36 views

Show that the closure of S in Y coincides with $\bar S \cap Y$

As a novice, it is hard to prove the following problem. Let Y be a subspace of a metric space X and let S be a subset of Y. Show that the closure of S in Y coincides with $\bar S \cap Y$, where $\bar ...
0
votes
2answers
46 views

A open subset of $\Bbb R$

Given the definitions in Open Subsets of open sets I need to prove that $\{x \in \Bbb R : |x|>2\}$ is open in $(\Bbb R , d_E)$ This seems to be true, however I don't know how to prove it without ...
2
votes
1answer
27 views

Continuous map of a compact set

Claim: If $f:X \to Y$ is continuous, where $X$ is compact, and $Y$ is Hausdorff, then $f$ is a closed map. Proof: Take $A \subset X$ to be closed in $X$. Now as $X$ is compact and by choice of $A$ we ...
3
votes
1answer
38 views

Munkres: Compact subsets of Hausdorff Space

Claim:If $A,B$ are compact disjoint subsets of the Hausdorff space $X$, then there exists disjoint open sets $U,V$ containing $A,B$ resp. Would I be on the right track in saying that since $A,B$ are ...
4
votes
1answer
59 views

Metric space in Topology class

On the set of integers $\mathbb Z$, show that the function d, defined as follow, is a metric : $$ d(x,y) = \begin{cases} 0 & \text{if } x=y \\ \min\{1/n! \mid n! \text{ divides } |x-y|\} & ...
0
votes
2answers
77 views

Discontinuity of the characteristic function

Let $A \subseteq \mathbb{R}^n$. Let $f(x) = \chi_A $ be the characteristic function, and put $D = \{ x : f(x) \; \; \text{is discontinuous} \} $. Then $\partial A = D $. MY try: Let $y \in D $. ...
3
votes
1answer
53 views

Path-Connectedness of Union

Can I have a hint to prove that $A \cup B$ is not path-connected, where $A = \{(x,y):0 \le x \le 1, y = x/n \text{ for n} \in \mathbb{N} \}$ and $B = \{(x,y):1/2 \le x \le 1, y = 0 \}$?
3
votes
2answers
43 views

Proof that a given projection map restricted to a subset is closed.

$\pi_{1}:\mathbb{R}^2\rightarrow\mathbb{R}, (x,y)\mapsto x$ is a projection map from $\mathbb{R}^2$ with the standard eulcidean topology, $\mathscr{T}_E$ to $\mathbb{R}$ with it's usual euclidean ...
1
vote
1answer
42 views

Closed subsets of compact sets are compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (1) Prove this using the definition of compactness. Can somebody prove it? I think we should select a open cover of S ...
0
votes
1answer
42 views

Prove that — the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$

Prove that the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$.
0
votes
2answers
50 views

About Hilbert spaces

How can I prove this fact: We're working in a Hilbert space $$ \mathcal{H} := \left\{ (x_n)_{n \in \mathbb{N}} \in {\mathbb{R}}^{\mathbb{N}} \mid \sum_{n=1}^{\infty}\,(x_n)^2 < \infty \right\} $$ ...
0
votes
1answer
37 views

Topological contraction on compact spaces

This is a follow up question. You can see the original here. I have the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a ...
1
vote
1answer
49 views

Intermediate Value theorem, $nth$ root function and continuity

So this problem is.. ridiculous to be honest. I have no idea where to start or what to do. Any help is appreciated. For the record, I am using the metric spaces definition of continuity.
1
vote
2answers
47 views

a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
0
votes
3answers
51 views

Discrete and compact subset must be finite

Show that a discrete and compact subset $D \subset \mathbb{C}$ must be finite. Does this conclusion hold if $D$ is just discrete and bounded? How about discrete and closed? Compact is the usual (for ...
1
vote
2answers
41 views

Every function from a discrete subset is continuous

Let $D \subset \mathbb{C}$ be a discrete subset and let $f : D \mapsto \mathbb{C}$ be a function. Show that $f$ is continuous. What's the best way to do this? I was thinking a proof by contradiction ...
0
votes
1answer
67 views

Discrete subsets in real analysis

So this is the first time I have seen the definition for a "discrete" subset. I want to first make sure that my understanding is correct. If $D$ is discrete, then D is essentially just the boundary ...
5
votes
1answer
168 views

Diagonal contained in interior of inverse image of open sets containing the diagonal implies continuity

Let $X, Y$ be topological spaces and let $f:X \rightarrow Y$ be a function and let $g = f \times f : X \times X \rightarrow Y \times Y$. I want to show that if: 1) $Y$ is normal and 2) for all ...
4
votes
2answers
90 views

Contraction of compact sets

I am trying to solve the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a non-empty set $A\subset X$ such that $f(A)=A$. ...
0
votes
1answer
22 views

Topology - closures of intervals with the order topology

a question from my h.w.: Let $(X,<)$ be a totally ordered set. Lets examine $X$ with the order topology. a. Prove that $\overline{(a,b)} \subseteq [a,b]$ (where the overline denotes ...