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; ...

learn more… | top users | synonyms (3)

3
votes
2answers
139 views

Two equal functions on a topological space

Can anybody help please help me, I have to answer this problem in topology: "Let $f$ and $g$ be continuous functions from the topological space $T$ into $\mathbb{R}$, with the usual topology. Show ...
3
votes
1answer
154 views

Question on the proof of a subspace of Polish space is Polish, iff it's a $G_\delta$ set.

Suppose, $X$ is a Polish space, $Y$ is a Polish subspace of $X$. $\{U_n\}_{n \in \Bbb N}$ is a basis of open sets of $X$. Let $A = \{ x \in \overline {Y} : \forall \epsilon \exists {n}(x \in U_n ...
3
votes
1answer
70 views

Isometries of [0,1] with natural topology

I'm trying to prove that metric space $([0,1], d)$ with natural topology has only one isometry other that identity. The hint in my book says that I should consider $f: [0,1] \rightarrow [0,1]$ such ...
3
votes
1answer
85 views

Prove that a connected space cannot have more than one dispersion points.

Prove that a connected space cannot have more than one dispersion points. I have no idea at all how can I able to tackle the problem.can anyone help me please. Also can I get some examples of ...
3
votes
2answers
257 views

Dieudonné complete and topologically complete are equivalent for every space $X$.

How can we show that: For every topological space $X$ the following conditions are equivalent: A space $X$ is topologically complete if $X$ is homeomorphic to a closed subspace of a product ...
3
votes
1answer
180 views

Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology?

My question is: Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology? How do you prove this? Do you use the countable open covering or do you use the accumulation point ...
3
votes
1answer
153 views

Is every $T_4$ topological space divisible?

A topological space $(X,\mathcal T)$ is said to be divisible iff for each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a symmetric neighborhood $V$ of the ...
3
votes
1answer
80 views

Topological vector space locally convex

Hello how to show the following: Let $X$ be a vector space. Then a norm on $X$ induces a topology under which $X$ is a locally convex topological vector space. Moreover, let $D$ be a nonempty ...
3
votes
1answer
2k views

homeomorphism between the real projective line and a circle

I'm currently following an introductory course in geometry and it was mentioned that the real projective line is homeomorphic to a circle. Could someone please state the topologies on both the real ...
3
votes
1answer
157 views

Closure of open set in a dense subspace of topological space.

Let $Y$ is a dense subspace of topological space $X$ and $U \mathop \subset \limits^{open} Y$. My purpose is to show that $Cl_{Y}(U)= Cl_{X}(V)\cap Y$. and i have two Question; Is it always true ...
3
votes
1answer
306 views

Topological properties of symmetric positive definite matrices

Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true? (a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$. (b) $S$ is ...
3
votes
1answer
703 views

Prove that the orbit of an iterated rotation of 0 (by (A)(Pi), A irrational) around a circle centered at the origin is dense in the circle.

I think the title of the question says it all. I unfortunately did not seem to conclude anything. Some ideas I had: It is easy to show that (given $T$ is the rotation) $\{T^n(\theta)\}$ is a set of ...
3
votes
1answer
72 views

How to show any convergent sequence is strongly discrete in Hausdorff space?

Given a space $X$ and $C \subset X$, we say that $C$ is strongly discrete if there exists a disjoint family $\{U_x: x\in C\} $ of open sets in $X$ such that $x\in U_x$ for all $x\in C$. The question ...
3
votes
3answers
171 views

What is the negation of this statement?

Let $(K_n)$ be a sequence of sets. What is the negation of the following statement? For all $U$ open containing $x$, $U \cap K_n \neq \emptyset$ for all but finitely many $n$.
3
votes
1answer
774 views

Continuous function on metric space

I'm trying to show: Let $(X,d)$ be a metric space and let $A, B$ be nonempty subsets, which are also closed and disjoint. Let $\rho_A:X\to \mathbb{R}$ be such that $\rho_A=d(x,A)$ and $\rho_B:X\to ...
3
votes
2answers
2k views

Cantor set: Lebesgue measure and uncountability

I have to prove two things. First is that the Cantor set has a lebesgue measure of 0. If we regard the supersets $C_n$, where $C_0 = [0,1]$, $C_1 = [0,\frac{1}{3}] \cup [\frac{2}{3},1]$ and so on. ...
3
votes
2answers
2k views

Prove that B is a basis for a topology

Let $X = \{0,1,2,3,\ldots\}$ (the non-negative integers), let $$B_1 = \{\{n\} : n \in X \text{ and }n > 0\}= \{\{1\}, \{2\}, \{3\},\ldots\}$$ $$B_2 = \{Z \subset X : X \setminus Z = \{1,2,\ldots ...
3
votes
1answer
613 views

Nowhere differentiability of Space-filling curves?

In a homework assigment, we were given a certain recursive definition of a space-filling curve $f : [0,1] \mapsto [0,1]^2$ and asked to determine where it is differentiable. My intuition tells me that ...
3
votes
2answers
404 views

Clopen subsets of the Cantor set.

In our Topology course, we have been studying the anti-equivalance of categories between zero-dimensional compact Hausdorff spaces and Boolean algebras. The Cantor set has come up a lot, and I have a ...
3
votes
4answers
631 views

Diagonal $\Delta = \{x \times x : x \in X \}$ closed in $X \times X$ implies that $X$ is Hausdorff [duplicate]

I think I have solved a problem in Topology by Munkres, but there is a small detail that is bugging me. The problem is stated in this question's title. I will write down the proof and will highlight ...
3
votes
1answer
2k views

Applications of Weierstrass Theorem & Stone Weierstrass Theorem

Problem: Prove that if $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ is a continuous function such that: $\int_{0}^{1}f(x)e^{nx}dx=0$ for all $n=0,1,2,...$. Prove that $f(x)=0$ for all $0\leqslant ...
3
votes
2answers
191 views

liminf in terms of the point-to-set distance

Let $\mathcal{X}$ be a normed space and $C\subseteq \mathcal{X}$. We define the point-to-set distance for the set $C$ to be: $$ d_C:\mathcal{X}\ni x \mapsto d_c(x):= \inf_{y\in C}\|x-y\| \in ...
3
votes
2answers
1k views

A metric space in which every infinite set has a limit point is separable

I am struggling with one problem. I need to show that if $X$ is a metric space in which every infinite subset has a limit point then $X$ is separable (has countable dense subset in other words). I am ...
3
votes
2answers
510 views

product and box topology

Let $f\:A\rightarrow\Pi _{\alpha\in J} X_\alpha$ be given by the equation $f(a)=(f_\alpha (a))_{\alpha \in J}$ where $f_{\alpha}:A\rightarrow X_\alpha$ for each $\alpha$. Let $\Pi X_\alpha$ have the ...
3
votes
1answer
111 views

Compact connected spaces have non- cut points

Let $X$ be a compact connected Hausdorff space with more than one point. Prove that there is point $x \in X$ s.t. $X \setminus \{x\}$ is connected.
3
votes
3answers
1k views

Exercises about Hausdorff spaces

two problems from Dugundji's book page $156$. (I don't know why the system deletes the word hi in this sentence) $1$. Let $X$ be a Hausdorff space. Show that: a) $\bigcap \{F: x \in F , F \ ...
2
votes
2answers
131 views

Give examples of compact spaces $A,B$ such that $A\cap B$ is not compact

If a topological space is Hausdorff then arbitrary intersection of compact sets is compact. How to find examples of compact subsets $A,B$ of a topological space $X$ such that $A\cap B$ is not ...
2
votes
1answer
81 views

Disjoint closed sets in a second countable zero-dimensional space can be separated by a clopen set

I want to prove the following: Let $X$ be second countable zero-dimensional space. If $A,B \subseteq X$ are disjoint closed sets, there exist is a clopen set $C$ such that $A\subseteq C$ and ...
2
votes
1answer
80 views

Basic problem on topology $( James Dugundji)$

Consider $(X, \tau)$ be a topological space . Then $Fr[Fr\{Fr(A) \}] = Fr[Fr(A)]$, where $Fr(A) =\overline A \cap \overline{A^c}$ is the frontier of the set $A$ Assume that $Fr(A) ...
2
votes
3answers
110 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of $\mathbb{R}$ and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
2
votes
1answer
124 views

Characterization of ideals of algebra of continuous functions on a compact space.

I was reading this planetmath page on the connections between the topology on a compact Hausdorff topological space $X$ and the maximal ideals on the algebra of continuous functions $C(X)$ on $X$, ...
2
votes
1answer
80 views

Proving the derived set $E'$ is closed.

I was reading the proof in Rudin, but it uses the metric. Is this not true if $X$ is a general topological space and $E' \subset X$ (especially if it is not Hausdorff $T_1$)? I can't come up with a ...
2
votes
3answers
134 views

Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis

I am not able to fill the gap in proof of following theorem which is stated as... Let $U$ be an open set in a locally compact hausdorff space $X$, $K\subset U$ and K is compact. Then there exists an ...
2
votes
1answer
59 views

Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?

Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and ...
2
votes
2answers
69 views

Show that a space is separable.

I want to show that the topological space $[0,1]^{[0,1]}$Separable space. That seems like a very complicated space...it is not countable, right? I have to find a set which is dense in the ...
2
votes
1answer
93 views

Polygonally connected

Prove that any open connected set in $C[0,1]$ is polygonally connected. (Here $C[0,1]$ is the space of real valued continuous functions on $[0,1]$ with the metric: $d(f,g)=$ $sup${$|f(x)-g(x)| : ...
2
votes
0answers
107 views

$\ f \colon X \to X $ ,continuous function where X is compact,Hausdorff space.Show $\exists A$ st $f(A) =A$.

Suppose $\ f \colon X \to X $ is a continuous function from a compact,Hausdorff space to itself. Prove that there exists a subspace $A$ such that $f(A) =A$. I came up with an answer based on nets ...
2
votes
1answer
417 views

Definition of a nowhere dense set

I'm currently studying metric spaces through Gamelin and Greene's Introduction to Topology. While studying about completeness I got stuck with this concept of nowhere dense subset. The book defines a ...
2
votes
1answer
67 views

Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is contained in a countable union of compact sets Does Baire space satisfy this ...
2
votes
1answer
723 views

Show simplicial complex is Hausdorff

I have a simplicial complex $K$ and I need to show that its topological realisation $|K|$ is Hausdorff. And $K$ need not be finite. I have very little idea on how to get started on this. Only that if ...
2
votes
1answer
127 views

Prove that the countable complement topology is not meta compact?

I have seen the proof of (countable complement topology is not meta compact) , which says that the countable intersection of open sets is open and thus uncountable, so this topology cannot be meta ...
2
votes
2answers
149 views

Proving that a particular restriction of a projection is a quotient map

I was hoping somebody could help me with the following problem: Let $\pi: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be the projection onto the first coordinate, and let $p=\pi|_X$, ...
2
votes
1answer
189 views

Is cantor set homeomorphic to the unit interval?

Can anyone help me with this question? Is cantor set homeomorphic to the unit interval? I (think that I) can see that there is an $f: C \rightarrow [0,1]_{inf}$ which is a surjective bijection ...
2
votes
1answer
138 views

Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes

What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open ...
2
votes
1answer
66 views

Topology Homeomorphism

Let $R$ be endowed with the subspace topology inherited from the euclidean topology. Let $A$ and $B$ be two subsets of $R$, where $A=\{3\}∪[4,5]∪\{6\}$ and $B=[3,4]∪\{5\}∪\{6\}.$ How do you show that ...
2
votes
2answers
433 views

For a compact covering space, the fibres of the covering map are finite.

I am stuck on the following exercise: Let $Y$ be a compact topological space, and $p:\ Y\ \longrightarrow\ X$ a covering map. Show that for every $x\in X$ the fibre $p^{-1}(x)$ is finite. Any ...
2
votes
1answer
638 views

Continuous images of Cauchy sequences are not necessarily Cauchy

Could you please provide an example for two metric spaces $X,Y$, a continuous function $f$ that maps $X$ to $Y$ and a Cauchy sequence in $X$, which is not mapped to a Cauchy sequence in $Y$ by $f$? ...
2
votes
1answer
178 views

How to prove the space of orbits is a Hausdorff space

Let $M$ be a connected smooth n-dimensional manifold and $G$ a lie group acting smoothly on $M$. for $x\in M$, the orbit $G\cdot x=\{g(x)\mid g\in G\} $ is a sub-manifold of $M$ and if the action is ...
2
votes
1answer
126 views

Is a uniquely geodesic space contractible? I

Is a uniquely geodesic space contractible ? We assume in addition that closed metric balls are compact. A post without this extra assumption is here.
2
votes
2answers
2k views

Prove that a compact metric space is complete.

I'm reading Intro to Topology by Mendelson. I'm in the section titled "Compact Metric Spaces". The problem is in the title. My attempt at the proof is as follows: Let $\{a_n\}_{n=1}^\infty$ be a ...