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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Some questions concerning continuity and relations

A lot of equivalent conditions for functions between topological spaces $$ X\overset f\longrightarrow Y $$ is proved on this site. Here some of them formulated from the perspective of 'relations': ...
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22 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 ...
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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 ...
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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 ...
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1answer
34 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?
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A topological function with only removable discontinuities

I've posted similar questions here and here, but no one has answered them to my satisfaction. Suppose that $f:\mathbb{R} \to \mathbb{R}$ is such that $\lim_{y\to x}f(y)$ exists for all $x$, that is, ...
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1answer
49 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?
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12 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 ...
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Is the function $ f(x,y)=xy/(x^{2}+y^{2})$ where f(0,0) is defined to be 0 continuous?

Is the function $ f(x,y)=xy/ (x^{2}+y^{2})$ where $f(0,0)$ is defined to be $0$ continuous? I don't think it is and I am trying to either show this by the definition or by showing that maybe a close ...
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1answer
24 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 $(0,1)$.
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27 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 ...
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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 ...
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2answers
96 views

Parametrization of $S^3$ embedded in $\mathbb R^4$?

I would like to know of any parametrization of the standard 3-sphere: {$(x_1,x_2,x_3,x_4): x_1^2+x_2^2+x_3^2+x_4^2=1$} embedded in $\mathbb R^4$. I know of parametrizations for $S^1$, for $S^2$ , ...
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1answer
42 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 $V\in \mathscr{T}$, ...
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1answer
52 views

Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
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1answer
30 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) ...
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87 views

Topological spaces vs. metric spaces

Are there any "realistic" examples of topological spaces that are not metric spaces. You are free to invent your own definition of "realistic". But, at a minimum, a realistic example is one that ...
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21 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!
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1answer
25 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 ...
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1answer
42 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 ...
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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 ...
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47 views

Example of locally metrizable Lindelöf Hausdorff space that is not metrizable

The title says it all. I've proved that if such space is regular, then it is metrizable. The proof relies on the ability to find Lindelöf neighborhoods inside metrizable ones, so any counterexample ...
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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 $\overline{A} \cap \overline{X-A}$. ...
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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 $[0,1]-A$. ...
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1answer
35 views

Fundamental questions on fundamental classes

Background: Hatcher's algebraic topology book. I would like to know whether, being given a closed orientable $n$-manifold with $\mathbf{Z}$-fundamental class $[M]\in H_n(M)$, there is a relation ...
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1answer
676 views

When is the vector space of continuous functions on a compact Hausdorff space finite dimensional?

I know that the vector space of all real valued continuous functions on a compact Hausdorff space can be infinite dimensional. When will it be finite dimensional? And how will I identify that vector ...
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27 views

Strange question about magnetic dipole in a plane at infinite distance

Please allow me to ask something rather unusual and perhaps completely naive. Suppose I have an electric current in a circular loop in a plane. Consider it just in a mathematical sense. The loop has ...
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1answer
59 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 ...
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1answer
48 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 ...
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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 ...
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46 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: ...
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1answer
44 views

Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
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1answer
82 views

Convex interior topology

I have found a fascinating example of topology on a vector space $V$, but I cannot prove its interesting properties to myself. Let $\mathcal{B}$ be the family of all convex symmetric (i.e. $\forall ...
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1answer
64 views

Isomorphism of Fundamental Groups (arcwise connected)

In an arcwise connected topological space $X$, we can show that the two groups $\pi(X,x)$ and $\pi(X,y)$ are isomorphic for $x,y \in X$ by defining a mapping $u: \pi(X,x) \to \pi(X,y)$ by $\alpha ...
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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 ...
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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 ...
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Proof verification: Munkres exercise 10, section 152

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 $\{X_\alpha\}_{\alpha ...
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$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 ...
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1answer
58 views

Sum of Neighborhoods of Zero

When do two neighborhoods of zero over a topological vector space add up as: $$aN+bN=(a+b)N\quad a,b\geq 0$$ I could imagine something like balanced might suffice... The problem is that I'd like to ...
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92 views

The topology generated by open intervals of rational numbers

Let $B = \{ \mathbb{R} \} \cup \{ (a,b) \cap\mathbb {Q} \ ,\ a\lt b \ ,\ a,b \in\mathbb{Q}\}$ Thus, a set $V \in B$ if it is either equal to $\mathbb{R}$ or if it is in the intersection of ...
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1answer
273 views

What distinguishes topological spaces from graphs?

Topology would not "work" if one reverted the "direction" in the definition of continuous maps $f$: $$\text{open}(x) \rightarrow \text{open}(f(x))$$ It has to be $$\text{open}(f(x)) \rightarrow ...
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Connectedness and non-local-connectedness of a subspace of $\mathbb R^2$

Let $(X,\tau)$ be the subspace of $\mathbb R^2$ consisting of the points in the line segments joining $(0,1)$ to $(0,0)$ and to all the points $(1/n,0)$, $n=1,2,3,\ldots$. Show that $(X,\tau)$ ...
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140 views

Is there a structure theorem for nonempty, compact, nowhere dense subsets of the real line? [closed]

Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line. Is there a theorem that describes the form of the elements of $X$? Context For open subsets of the line, such a ...
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1answer
49 views

Motivation for the proof of the associativity of multiplication of equivalence classes of paths

After having defined the equivalence classes of paths in a topological space in chapter two of the book A Basic Course in Algebraic Topology, William S. Massey proves the lemma The multiplication ...
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How can I show that that compact subsets of $\mathbb{R}$ in this topology are finite subsets.

Let $\tau$ be the topology on $\mathbb{R}$ which has as base the collection of all sets of the form $O \setminus C$ where $O \subset \mathbb{R}$ is an open set in the standard topolgy of ...
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1answer
50 views

Question about an example of Quotient space.

I am new to the concept of Quotient space, and I have an example of a Quotient space from one of my lecture notes, which I can't understand. Here is the example quoted from the lecture note: A ...
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1answer
47 views

The number of connected components of $SL(2, \mathbb{R})$ which keep $x^2 - y^2$ invariant

Working on yet another past comprehensive exam question. Let $S$ be the set of real $2\times 2$ matrices with determinant $1$, keeping invariant the form $x^2 - y^2$. Regard $S$ as a subset of ...
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A proof that continuity of $f:X\to Y$ is equivalent to $\overline{f^{-1}(M)}\subset f^{-1}(\overline{M})$.

Given two topological spaces $\left\langle X,\tau\right\rangle $, $\left\langle Y,\sigma\right\rangle$ and a function $X\overset{f}\longrightarrow Y$. Would someone please sketch a proof that (1) ...
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1answer
31 views

What is the topology generated by a basis?

Reading Munkres' text on Topology, we get the fairly straight-forward definition of a basis: Blabla $\mathcal{B}$ is a basis for a topology on $X$ if $\mathcal{B}$ is a collection of subsets of ...
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71 views

How many subspace topologies of $\mathbb{R}$?

Say two subsets of $\mathbb{R}$ are equivalent if they are homeomorphic, with the subspace topology. How many equivalence classes are there? It's immediate that there are at least $\beth_0$ (we can ...