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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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
62 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
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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 ...
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1answer
60 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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+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 ...
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2answers
86 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 ...
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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 ...
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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 ...
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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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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 ...
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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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2answers
35 views

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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2answers
42 views

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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0answers
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
51 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 ...
3
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1answer
59 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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4answers
57 views

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
32 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 ...
4
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1answer
78 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 ...
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1answer
44 views

The topology a line inherits as a subspace of $\mathbb{R}_l \times \mathbb{R}$, or of $\mathbb{R}_l \times \mathbb{R}_l$ (Munkres)

I have a question about Exercise 8, Chapter 2, Pag. 92, from the Munkres's book: If $L$ is a straight line in the plane, describe the topology $L$ inherits as a subspace of $\mathbb{R}_l \times ...
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0answers
36 views

Weak Topology Problem [closed]

Let $C(X,Y)$ be the set of all continuous functions between a locally compact topological space $X$ and an arbitrary topological space $Y$. We define the weak topology on $C(X,Y)$ by taking as ...
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2answers
44 views

Prove that $Y$ which has at least cardinality of $\mathbb{R}$, is path connected

Let $X$ be an infinite set and $\tau=\lbrace A\subset X : A = \emptyset \quad \text{or} \quad X \setminus A \quad \text{is finite}\rbrace$. Let $Y$ be a subspace of the topological space ...
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2answers
95 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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1answer
130 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
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1answer
57 views

Regarding open subsets in topology

(Probably due to my lack of experience with the subject, I see that my question is horribly written. If you are to answer, a beginner-friendly explanation of the basis of a topology and the topology ...
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3answers
37 views

Local compactness is preserved under continuous open onto mappings

If $f$ is a continuous open mapping of a locally compact space $(X,\tau)$ onto a topological space $(Y,\tau_1)$, show that $(Y,\tau_1)$ is locally compact. The definition of locally compact is ...
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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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1answer
51 views

A question from Otto Forster's book

I'm tackling exercise 8.2 on page 59 which goes as follows: Let $X$ and $Y$ be compact Riemann surfaces, $a_1,\dots,a_n\in X$ and $b_1,\dots,b_m\in Y$ distinct points and ...
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1answer
48 views

Topology, Proof of function being continuous

Let $ (X_i,d_i),(Y_i,d_i^*)$, $i=1,\ldots,n $ be metric spaces. Let $ f_i:X_i \to Y_i, i=1,...,n $ be continuous functions. Let $$ X = \prod_{i=1}^{n} X_i , Y = \prod_{i=1}^{n} Y_i $$ and ...
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3answers
176 views

Klein bottle as two Möbius strips.

I read that glueing together two Möbius strips along their edges creates a surface that is equivalent to the so-called Klein bottle. The Möbius strip comes in two versions that are mirrored ...
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3answers
61 views

Topological properties of the so-called “plane at infinity”.

When 3D-Euclidean geometry is extended with ideal points at infinity, a whole "plane at infinity" is added to the geometry. Apart from metric properties it has become a 3D projective space and the ...
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18 views

Each infinite subspace of a KC-space …

A space in which all compact subsets are closed is called KC-space. A space in which every infinite set contains an infinite subset with only a finite number of accumulation points is said to have ...
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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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2answers
37 views

“Openness” of sets in $\tau$, where $\tau$ is generated by a basis $\mathcal{B}$

I am learning topology by working through the book written by Munkres. According to Munkres, $\mathcal{B}$ forms a basis on $X$ if: $\forall x \in X, \exists B \in \mathcal{B} : x \in B$ ...
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2answers
53 views

Deciding if sets are bounded and/or closed

How do we find out if a given set is bounded or closed? 1) $\{(x,y,z)\in \mathbb R^3 : x^2+2y^2-3z^3=1\}$ 2) $\{(x,y,z) \in \mathbb R^3 : |x|+2|y|+3|z|\le 1\}$ 3) $\{(x,y)\in \mathbb R^2 : ...
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1answer
36 views

Connected subsets of metric (or T1) spaces

I have proved some statements about connected subsets of a metric space. They are really basic, but I want to make sure that they are true. Would someone please tell me whether these statements are ...
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1answer
23 views

One-point sets are G$_\delta$ in first-countable $T_1$ spaces

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere. I only need help with my proof in particular. Show that in a ...
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1answer
53 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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2answers
39 views

Boundary, closure, and interior of $\{(x,y)\in \mathbb{R}^2 \;|\; x \in \mathbb{Q} \text{ and }y>0\} \subseteq \mathbb{R}^2$

Find the closure, boundary, and interior of the following subset of $\mathbb{R}^2$: $$ A=\{(x,y)\in \mathbb{R}^2 \;|\; x \in \mathbb{Q} \text{ and }y>0\} $$ It is clear that the closure is ...
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1answer
35 views

Infinite topological space with cofinite topology is not Hausdorff

I found a proof to the question, but mine is completely different (sort of). Is this correct? If $X$ were Hausdorff, then consider $u,v \in X$ with disjoint neighbourhoods $U, V$ that separates ...
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315 views

An open interval is an open set?

First, the question I have is very similar to this question, but I hope it doesn't get closed as duplicate since I'm stuck nevertheless. I'm trying to algebraically prove that an open interval is an ...
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1answer
34 views

Can we always form a closed set from a given domain

I have certain doubts regarding open and closed sets. So, anyone please help me. Can we always form a closed set from a given domain (open connected subset of the complex plane), i.e. choosing some ...
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1answer
36 views

Problem on product space of Sorgenfrey line.

Let $(\mathbb R,\tau)$ be Sorgenfrey line, $(\mathbb R^2,\tau_1):=(\mathbb R,\tau)\times (\mathbb R,\tau)$. Let $L = \{(x, y) : x, y\in\mathbb R^2, x + y = 0\}$. Show that the line L is closed in ...
2
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1answer
33 views

A countable, compact KC-subspace of a hereditarily Lindelöf minimal KC-space

A space in which all compact subsets are closed is called KC-space. A space in which every infinite set contains an infinite subset with only a finite number of accumulation points is said to have ...
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1answer
24 views

Show that the product of Sorgenfrey line is regular.

Let $(\mathbb R,\tau)$ be Sorgenfrey line, show that $(\mathbb R^2,\tau_1):=(\mathbb R,\tau)\times (\mathbb R,\tau)$ is regular. I am struggling in this problem, and haven't made much progress. ...
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1answer
40 views

Can every open set of $\mathbb{R}$ be written as countable union of disjoint open bounded intervals?

I know that the following two statements are correct. Every open set of $\mathbb{R}$ be written as countable union of disjoint open intervals ( including open rays and $\mathbb{R}$). Every open ...
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2answers
38 views

questions about the closed graph of topological curve?

Suppose that $M$ is a topological space,$\gamma:dom(\gamma)\rightarrow M$ is continuous,where $dom(\gamma)\subset \mathbb{R}$ is a open interval. Is it possilble that there exists special topological ...
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3answers
66 views

If $(X,\tau)^n$ is Hausdorf, is $(X,\tau)$ also Hausdorff?

If $(X,\tau)^n$ is Hausdorf, is $(X,\tau)$ also Hausdorff? I know that product of Hausdorff space is Hausdorff, but I want to know if this weaker converse of it is true. Thanks.
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1answer
41 views

What is a “closed subspace” of a topological space?

I was reading a proof online and it linked to a book by Munkres which says Every closed subspace of a compact space is compact. I dug out the book and searched the index for this term. ...