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 (2)

1
vote
2answers
19 views

Homology group of 3-fold sum of projective planes

I want to calculate the homology group of the 3-fold sum of projective planes defined by the labelling scheme $aabbcc$. For this I will use the following corollary from Munkres: Corollary 75.2: Let ...
0
votes
1answer
27 views

Homology group of space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$

I have to calculate the homology group of the quotient space $X$ given by the labelling scheme $aabcb^{-1}c^{-1}$ and then determine to which of the following spaces it is homeomorphic: $S^2, ...
0
votes
1answer
19 views

Arbitrary Fundamental Group and Surfaces

someone had explained to me how to construct arbitrary space $X_G$ such that $\pi_1(X_G) \cong G$, but i don't remember the end. The idea was the following : take a presentation of the group, and ...
0
votes
1answer
23 views

The set of all limit points $A'$ of a subset of a topological space $X$ is empty if $\tau = 2^X$

Proposition: If $X$ is a topological space with $\tau = 2^X$, then $A' = \emptyset$ where $A \subset X$ I found the proof and it uses the fact that if $x \in A$, then $\{ x\} \cap A - \{x \} = ...
-2
votes
3answers
60 views

I neet to prove that the set $ A:=\{ \frac {1}{n} | n \in \mathbb{N}\}\bigcup\{ 0\}$ is closed in R.

$ A:=\{ \frac {1}{n} | n \in \mathbb{N}\}\bigcup\{ 0\}$ is a closed set in $\mathbb{R}$ by the definiton. I can't use that $cl(A)=A$ iff a is a closed set.
1
vote
2answers
36 views

Closed sets in product topology

I have an assignment, I have to proof that arbitrary product of close sets is closed in the product topology, I think I have to use complements and treat with opens, what do you think?
1
vote
1answer
37 views

Space of Functions: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
1
vote
1answer
23 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
2
votes
0answers
34 views

Basic question about lifting maps to covering spaces

Any continuous map $f: X_1 \to X_2$ "lifts" to a map $\tilde f: \tilde X_1 \to \tilde X_2$ (provided that $X_1$ and $X_2$ have universal covers). The space $\tilde X_1$ is certainly ...
1
vote
1answer
75 views

Is true the boundary of compact set of $\mathbb{R}^n$ have Measure Zero?

Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \rightarrow [0, \infty[$ a measurable function. Suppose that there exist $C>0$ such that $$\int_K f dm < C,\ \forall\ K\subset\Omega,\ K\ ...
1
vote
1answer
31 views

$X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point

Im trying to show that: for $X,Y$ topological spaces $X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point while $[X,Y]$ denote the set of homotopy classes of maps of $X$ ...
0
votes
2answers
64 views

How to show that every continuous function from $[0:1]$ to $[0:1]$ has a fixed point?

This exercise is from Munkres topology: Let $f:[0:1]\rightarrow [0:1]$ be a continuous function. How can we prove that there exists some point $x\in [0:1]$ such that, $f(x)=x$? Any ideas please?
0
votes
2answers
41 views

Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
0
votes
1answer
35 views

Compactness of the Grassmannian $G(k,n)$

Related to this question, suppose we define $G(k,n)$ to be the set of $n\times k$ matricies with rank $k$, equipped with the quotient topology of $\mathbb{R}^{nk}$ by the equivalence relaiton $$A\sim ...
1
vote
1answer
53 views

Operators on the family of all subsets of a topological space that maybe generates a base for these family.

I will try to do at least something of my first question. Given a topological space $\langle X,\tau\rangle$, we define two operators on $2^X = \{ A : A \subseteq X \}$ as follows. For $\alpha ...
2
votes
6answers
78 views

Rudin's Topological Definition of an Open Set — Does it Disagree with the Metric Space Definition?

I wanted to share this definition of an open set, which made me uncomfortable. It comes from Rudin's Real and Complex Analysis and begins with the definition of a topology: A collection $\tau$ of ...
1
vote
1answer
30 views

Homeomorphism of a Genus-2 Surface

Does there exist a homeomorphism from a genus-2 surface, the connected sum of 2 tori, to two circles, $S^1$, intersecting at a point? Intuitively it seems that the double torus can be squeezed into ...
1
vote
1answer
59 views

topology defined on the set $\mathbb{R}^\mathbb{R}$?

What is the topology defined on the set $\mathbb{R}^\mathbb{R}$ of functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that allows us to talk about convergence of sequences in $\mathbb{R}^\mathbb{R}$?
0
votes
1answer
58 views

Properties of preimages and intersections of sets

I am working through Bert Mendelson's "Introduction to Topology" and am having some trouble with proofs. The text in well presented but to get a proper understanding I am working through the ...
3
votes
2answers
40 views

Connected topological spaces, product is connected

Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is ...
2
votes
3answers
69 views

Openness of path connected components of open subsets of $\mathbb C$

Let $\Omega\subset \Bbb{C}$ be an open set. My textbook states that every path connected component of $\Omega$ is open. I can't seem to understand why that is. Why does every point have to contained ...
-3
votes
0answers
47 views

Solutions to Topology by Munkres [on hold]

I was searching for solution to chapters 1, 2 & 3 of the book 'Topology' by James Munkres. Any suggestions where it can be available.
0
votes
1answer
16 views

Strong approximation of operators.

If I want to approximate strongly an operator $T$ with another in a subset $A \in L(H)$ why is not enough to ask "for every $\epsilon>0$ there is an operator $S\in A$ such that for every $\eta \in ...
0
votes
1answer
34 views

Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
1
vote
0answers
37 views

Some questions concerning continuity and relations

A lot of equivalent conditions for functions between topological spaces $$ X\overset f\longrightarrow Y $$ are proved on this site. Here some of them formulated from the perspective of 'relations': ...
0
votes
1answer
26 views

Countable product of closed set is closed?

This is my problem: Let $X=\prod_{i=1}^\infty X_i$. Product of $C_i$ closed requires its complement open. i.e. $X\setminus\prod_{i=1}^\infty C_i=\prod_{i=1}^\infty X_i\setminus C_i$ open. But ...
1
vote
2answers
21 views

Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
0
votes
1answer
32 views

Interesting and intuitive affirmation involving convex sets

Let $\Omega_1$ and $\Omega_2$ two open, bounded and convex domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ and $0 \in \Omega_2.$ Suppose that for each $x_0 \in \partial (\Omega_1 ...
1
vote
1answer
25 views

Simple question about an exemple in covers

I don't get the last one (I underlined it in red ) take $(0,1)$(which is an unbounded subset of $\mathbb R$) then if we take $a=0$ then this set $\{(0,1)\}$ will cover the subset $(0,1)$.
-2
votes
1answer
37 views

metric spaces and topology [on hold]

Let $d_1,d_2$ be metrics on $X$ such that any sequence $(x_n)$ converges in $(X,d_1)$ iff it converges in $(X,d_2)$ to the same point. Must $(X,d_1)$ and $(X,d_2)$ have the same topology?
-2
votes
0answers
35 views

limit point and topology [on hold]

Can we define a topology on set of naturals $\mathbb N$ in which every point is a limit point? i know that set $\mathbb N$ has no limit point but can we define a topology on $\mathbb N$ such that ...
1
vote
1answer
43 views

Continuity definition and theorem in a topology

This is an extremely common theorem, I have a function $f$ that maps $f:(X,\mathscr{S})\to(Y,\mathscr{T})$. I want to show that $f$ is continuous if and only if for all $V\in \mathscr{T}$, ...
0
votes
1answer
54 views

Banach Spaces: Totally Bounded vs. Bounded

Are the finite dimensional Banach spaces precisely those ones in which subsets are totally bounded iff they're bounded?
0
votes
1answer
33 views

Non-punctual Boundary

In the book of Bill Thurston, Three dimensional geometry and topology, there is an exercise to show torus can be partitioned into 7 countries, each on one piece and has common (non-punctual) ...
1
vote
1answer
48 views

What kind of space is this: $\Bbb{R}^n\times\Bbb{S}_{++}^n$?

Let $\Bbb{R}^n$ be the Euclidean space of $n$-dimensional column vectors with real coefficients. Moreover, $\Bbb{S}_{++}^n$ be the space of symmetric positive definite $n\times n$ real matrices. We ...
2
votes
1answer
18 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
3
votes
2answers
24 views

Complement of a solid genus-2-handlebody in $S^3$

I'm not sure if this is a stupid question or not but is the complement of a solid genus-2-handlebody in $S^3$ also a solid genus-2-handlebody? Thanks!
2
votes
1answer
27 views

The tetrahedron is a topological manifold

I have been thinking that simplicial complexes can not be given topological manifold structure since a simplicial complex is a union of simplices of different dimensions, hence there may be are points ...
1
vote
2answers
45 views

The boundary of an open subset of $[0,1]$ containing all rationals in $(0,1)$

If $A\subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that each rational number of $(0,1)$ is contained in some $(a_i,b_i)$, prove that the boundary (frontier) of $A$ is $[0,1]-A$. ...
0
votes
0answers
38 views

Darboux Integrable Functions vs. Uniform Closure of Simple Functions

Is there a Darboux integrable function $f:[0,1]\to E$ with values in a Banach space $E$ that is not the uniform limit of simple functions $s=\sum_\alpha\chi_{A_\alpha}$ taken over finite sums with ...
-3
votes
1answer
60 views

How to show that $\mathbb R^n$ is an open set?

How to show that $\mathbb R^n$ is open using open rectangles? Open rectangle is defined as $$(a_1,b_1)\times \ldots \times(a_n,b_n)$$ I am really stuck; first time doing topology and this is just ...
6
votes
1answer
58 views

A locally metrizable, Lindelöf Hausdorff space that is not metrizable

I am looking for an example of locally metrizable, Lindelöf Hausdorff space that is not metrizable. I've proved that if such space is regular, then it is metrizable. The proof relies on the ability ...
2
votes
1answer
59 views

Topological and algebraic interiors

I read on a functional analysis book that in a normed, real or complex, space $V$ the algebraic interior of a set $S\subset V$ defined $J(S):=\{x\in S:\quad\forall y\in V\quad\exists ...
1
vote
0answers
57 views
+50

Proof verification: Munkres exercise 10, section 23

Can someone please verify my proof or offer suggestions for improvement? I'm thoroughly confused by this question, and I'm sure there's a mistake somewhere in my proof. Let $\{X_\alpha\}_{\alpha ...
3
votes
2answers
84 views

About the proof of the Heine-Borel Theorem

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have a question about the prove of theorem 3.3.4 on page 84 (i.e. the Heine-Borel theorem). To be more specific, let us ...
2
votes
0answers
66 views

$C(X)$ is finite dimensional iff $X$ is finite [duplicate]

If $X$ is compact Hausdorff space and $C(X)$ is the set of all continuous complex valued functions on $X$,then prove that $C(X)$ is finite dimensional if and only if $X$ is finite. My problem:If we ...
3
votes
2answers
33 views

The topology generated by a basis is the intersection of all topologies containing that basis.

This question is from Munkres' Topology, section 13, exercise 5. I ask for verification and/or comments upon mistakes and inaccuracies. Let $\mathcal{A}$ be a basis for a topology on $X$. We are ...
1
vote
1answer
19 views

The intersection of a connected subspace with the boundary of another subset

Can someone please verify my proof or offer suggestions for improvement? Definition/Notation: The boundary of $A$, denoted by $\operatorname{Bd}(A)$, equals $\overline{A} \cap \overline{X-A}$. ...
5
votes
2answers
83 views

A subset of $[0,1]\times[0,1]$ containing at most one point from each horizontal and vertical section whose boundary is $[0,1]\times[0,1]$

How can one build a subset $A\subset [0,1]\times[0,1]$ containing at the most one point from each horizontal and each vertical section and whose boundary (frontier) is $[0,1]\times[0,1]$? I don't ...
1
vote
1answer
40 views

“Broken-line paths” in $\mathbb R^n - \{ 0 \}$

In Munkres's Topology, he says: Suppose $x$ and $y$ are two different points from zero of the punctured euclidean space $\mathbb{R}^n -\{0\}$. We can join them a path by the straight-line path ...