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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Boundary of a ball

Show that the open ball $B(0,1) = \{(x,y): x^2 +y^2 < 1\}$ has the boundary $x^2+y^2=1$. I understand that the boundary is the closure of the ball minus the interior. So, if i can show that the ...
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2answers
70 views

Continuity in topology

I came across an exercise problem which says: Let $X=\{1,2,3\}$ with topology $T= \{\{1,2\}, \{1,2,3\}, \{3\}, \emptyset\}$ and $Y=\{1,2\}$ with topology $t=\{\{1\},\{2\},\{1,2\},\emptyset\}$. When ...
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1answer
57 views

Square metric as basis for $R^{n}$

i wish to show that open cubes centered at every point form a basis for $R^{n}.$ In this, i am trying to show than an open ball is a union of smaller open cubes, in which case is open, and hence the ...
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4answers
605 views

Is there a continuous bijection between an interval and a square: $[0,1] \mapsto [0,1] \times [0,1]$?

Is there a continuous bijection from $[0,1]$ onto $[0,1] \times [0,1]$? That is with $I=[0,1]$ and $S=[0,1] \times [0,1]$, is there a continuous bijection $$ f: I \to S? $$ I know there is a ...
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3answers
59 views

Topology generation

What does it mean for a topology to be generated? For example $X=\mathbb{R}$ be topology generated by $[a,b)$. Isn't the topology a collection of open sets? $[a,b)$ is not open though.
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2answers
78 views

$C(K)^{*}$ in the weak*-topology

It is clear that for a compact Hausdorff space $K$ the space of continuous complex (or real) functions $C(K)$ is a Banach space with the "sup" norm. What's not clear to me is whether there exists a ...
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1answer
111 views

$X$ Hausdorff space, $Y \subseteq X$ compact $\implies $ Y closed

Suppose $X$ is a hausdorff Space and $Y \subseteq X$ compact. Then $Y$ must be closed My Attempt: We need to find an open set $O$ such that $O \subseteq X \setminus Y$, then $Y$ is closed. Let ...
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1answer
81 views

Interior points in topology

Let $X= \mathbb{R^2}$ with subway metric. Here subway metric is the Paris metric. Let $A= [-1,1] \times \{0\}$. What is the interior point of $A$? I would say it is $(-1,1) \times \{0\}$ but I got it ...
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1answer
167 views

If S is a closed set, prove that $\partial S$=$\partial(\partial S)$

If S is a closed set, prove that $\partial S$=$\partial(\partial S)$. I'm trying to prove this using the equation $\partial S$=cl($\partial S$)=int($\partial S$)$\cup \partial(\partial S)$, then we ...
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1answer
88 views

Compact spaces where not all compact subsets are closed

A topological space $(X,\tau)$ is called $C-C$ iff the closed sets in $X$ coincide with the compact sets in $X$. A topological space is called a $US$-space provided that each convergent sequence has ...
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1answer
58 views

A topological space $C- C$

A topological space is called a $C- C$ space iff the closed sets in $X$ coincide with the compact sets in $X$. Do the two statements below hold? (1) : Let $(X,\tau)$ be a $C- C$ space and let ...
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2answers
548 views

Finite subsets of Hausdorff spaces

quick question. There is a question in the book saying "Every finite subset of a Hausdorff space is closed." The proof uses the fact that for each point in $X\setminus\{p\}$ there is a neighborhood ...
5
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1answer
76 views

Hausdorff spaces from continuous functions

The question is to prove a topological space is Hausdorff if for every $p$ in the space there exists a continuous function $f_{p}$ such that $f^{-1}(0) = \{p\}.$ (The inverse here is implied as ...
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1answer
257 views

topology: embedding

Billingsley writes in his book convergence of probability measures: " If $S$ can be embedded as an open set in a some complete metric space, then it is topologically complete". I have taken one ...
5
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1answer
133 views

Fréchet-Urysohn space

We know that if $p$ is the limit of a subsequence $(x_{n_{k}})$ of the sequence $(x_n)$ in $X$ then $p$ is a cluster point of the sequence. For sequential spaces it does not to hold that a cluster ...
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1answer
265 views

Open and closed sets in topology

I recently came across the following example which was really surprising: Let $X= [1,2] \cup [3,4]$? Here $[1,2]$ is both open and closed and same holds fo $[3,4]$. But how is it possible to ...
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1answer
30 views

Open set in point set topology

Let $X=\mathbb{R}$ with standard topology and $B=[1,2] \cup [3,4]$. Is $[1,2]$ open in $B$? The answer is yes because $[1,2] \cup [3,4] \cap (1/2, 5/2)$. I don't understand the answer and reasoning. ...
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1answer
101 views

Open set and basis in Topology

How is the following set open: An example of open set is $(-1,1) \times [-1,1]$. Prove that the given open set is a basis for $X= \mathbb{R}$ (with standard topology) $\times \mathbb{R}$ (with ...
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3answers
130 views

Are the following sets open subsets of $\mathbb{R}$

I need to determine whether $[2,4]$ and $\mathbb{R}-\mathbb{Q}$ are open subsets of $\Bbb R$. For $[2,4]$: I know for a subset to be open then $\forall x\in V$ $\exists (a,b)$ s.t. $x\in ...
4
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1answer
162 views

Defining a quotient manifold with gluing

I'm trying to find conditions on the gluing map between two manifolds so that the quotient space will be a smooth manifold, and the inclusion map will be a diffeomorphism. Specifically, Suppose ...
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2answers
97 views

Prove: A subset V of $\mathbb{R}$ is open iff V is equal to a union of open intervals

The proof of the theorem is given to me in the book but I need some clarification about specific aspects of the proof that the book thinks is trivial: $\Rightarrow$ Assume V is a open set of ...
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1answer
37 views

closed maps on topological spaces

Prove that $f$ is closed if and only if $f(\text{cl}(A)) \supseteq \text{cl}(f(A)).$ I can show the forward directions by saying: Suppose $f$ is closed. Then, $f(\text{cl}(A)) = f(A) \cup f(\partial ...
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1answer
310 views

Clarification on this corollary of the Arzela-Ascoli Theorem

I am given the following corollary without proof: A family of continuous functions on a compact metric space into $\mathbb R^m$ is compact iff it is closed, equicontinuous and bounded. Does ...
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1answer
67 views

Metric Space and ordered field

If we have an ordered field $ \mathbb{F} $, can we consider a natural metric involved with this space? What should be this metric? thanks
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2answers
86 views

Interior and closure of $n^3$ in the topology on $\mathbb Z$ generated by bi-infinite arithmetic progressions

Consider the topology on the integers $\mathbb{Z}$ defined by the collection of all bi-infinite arithmetic progressions $$\mathscr{B} = \{A_{a,d} \mid d\in \mathbb{N}, 0 \le a < d\}$$ $$A_{a,d} = ...
2
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1answer
47 views

Metric in an ordered field

Suppose that we have $ \mathbb{F} $ an ordered field with a metric d and $x,y \in \mathbb{F} $ non negative numbers. It is possible to affirm that if $ x \leq y $ then $d (x,0) \leq d(y,0) $? If we ...
2
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1answer
34 views

Distance between differential operators

Given two differential operators say $D_1$ and $D_2$ is there any meaningful way to define distance between them, does there exist some metric $d(D_1,D_2)$ that satisfies all the necessary properties? ...
3
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2answers
85 views

Connected, but it is not continuous at some point(s) of I

I have a mathematics problem, but I have no idea. please help me... The problem is "Give an example of a function $f(x)$ defined on an interval I whose graph is connected, but is is not continuous at ...
5
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1answer
104 views

Given metric space $X$ with no isolated points, and dense $Y \subset X$, find $Z \subset Y$ so that $Z$ and $Y-Z$ are both dense in $X$.

The question is: Given metric space $X$ with no isolated points, and dense $Y \subset X$, find $Z \subset Y$ so that $Z$ and $Y-Z$ are both dense in $X$. Now, if a metric space $X$ with no isolated ...
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1answer
35 views

Defining a continuous map via homeomorphisms

Let $X$ be a $T_{2}$ space and there exists in $X$ a sequence $E_{1},E_{2},E_{3},...$ of closed disjoint homeomorphic copies of $X$. I am trying to define a continuous map $f$ from $X$ into $X$ such ...
2
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1answer
211 views

free and uniform ultrafilter

An ultrafilter $\mathcal{F}$ is said to be free if $\cap \mathcal{F} = \emptyset$. An ultrafilter $\mathcal{F}$ is an uniform ultrafilter in $X$ if $|F| = |X|$ for every $F \in \mathcal{F}$. ...
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0answers
88 views

Let $\mathcal{F}$ be a filter on $X$. $\mathcal{B} \subseteq P( X)$ is called a filter- base…

Let $\mathcal{F}$ be a filter on $X$. $\mathcal{B} \subseteq P( X)$ is called a filter- base satisfies bellow conditions: ( 1 ) : ‎$ ‎\mathcal{B}‎ ‎\neq‎ ‎\emptyset‎ $‏ ‎‎( 2 ) : ‎$ ‎\emptyset ...
3
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1answer
671 views

Separable metric space has a countable base

A collection $\{V_{\alpha}\}$ of open subsets of $X$ is said to be a $\textit{base}$ for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have ...
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2answers
188 views

What are some examples of discontinuous functions from $\mathbb{R}^2$ to $\mathbb{R}^2$

If I have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $\mathbb{R}^2$ is equipped with the euclidean topology, in both cases, what are some examples of discontinuous functions?
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0answers
31 views

Transforming the measure in $CP^1$ mapping from Riemann sphere to $\mathbb{C}^2$-plane

I would like to know how the measure changes in $CP^1$ mapping from Riemann sphere (2-sphere) to $\mathbb{C}^2$-plane. Let a point on the 2-sphere is given by the vector ...
3
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1answer
148 views

Homeomorphism in $\mathbb{R}$ with the upper limit topology.

Consider $\mathbb{R}$ with the upper limit topology (open sets are of the form $(a,b]$) and consider the subsets $(0,1]$ and $(0, +\infty)$ with the corresponding relative topologies. Show that ...
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1answer
78 views

Under what conditions, the converse of the claim “continuous functions take limits to limits” is also true?

Let $X,Y$ be topological spaces and $f:X \rightarrow Y$ a function, and a set $A \subset X$ where $A = \{x_\lambda | \lambda \in \Lambda\}$ where $\Lambda$ is a directed set. Suppose that x is a limit ...
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1answer
68 views

Separability and open balls

I have a super basic question that for some reason has been eluding me for quite a while. This question actually came up in the context of weak convergence of probability measures on the space ...
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0answers
96 views

Condition for a compact set to be the support of a continuous function?

I am studying Rudin's Real and Complex Analysis exercises and I am currently thinking about the following: Is there a characterization of the class of compact sets of $\mathbb{R}$ which are ...
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0answers
71 views

What is the use of locally connected spaces?

One of the main properties of locally connected spaces is that their connected components are clopen and thus, they are homeomorphic to the colimit of their connected components. This is good to ...
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1answer
56 views

Is the continuous map between CW-complexes a cofibration?

If $f:A \rightarrow X$ is a continuous map between CW-complexes, then is $f$ necessarily a cofibration? I know that when $A$ is a subcomplex of $X$ and $f$ is the inclusion, the conclusion is true. ...
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2answers
51 views

counterexample about sets of limit points

Let $X$ be a topological space, and $A,B \subseteq X$. I know that $(A ∪ B)'$ may not be contained in $(A' ∪ B')$. But I don't know a specific example that show the above statement. Please let me ...
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0answers
82 views

Compact Hausdorff implies product of quotient map is a quotient map?

Let $X$ be compact Hausdorff and let $q : X \to Y$ be a quotient map. Is it true that $f : X \times X \to Y \times Y$ with $(x_1, x_2) \mapsto (q(x_1), q(x_2))$ is a quotient map?
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1answer
133 views

$\mathbb{RP}^3$ is homeomorphic to the solid ball with antipodal points identified

I am reading the book Application of Path integrals by Schulman, which has a chapter on applications of homotopy theory to path integrals. In that he says we can geometrically describe $SO(3)$ by a ...
2
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3answers
128 views

Continuity for topological spaces

After reading the definition of a continuous map on general topological spaces, my question is the following: Suppose $f$ is continuous from $\mathbb R$ to $\mathbb C$ given by $x \mapsto e^{ix}$. ...
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2answers
96 views

A set is called sequential - closed if it contains all its sequential limit point.

A set is called sequential-closed if it contains all its sequential limit point. A set is called sequential-open if it is a sequential neighborhood ($N$ is a sequential neighborhood if whenever $x_n$ ...
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1answer
121 views

Abstracting Completeness

Introduction This may seem like a weird question or even a silly one, but topology is vast and I find I make quicker progress working my way through it by trying ideas out loud within earshot of ...
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1answer
96 views

Is the union of finitely many open sets in an omega-cover contained within some member of the cover?

Let $\mathcal{U}$ be an open cover of $\mathbb{R}$ (Standard Topology) such that $\mathbb{R} \not \in \mathcal{U}$ and for any finite set $A$ there is a $U \in \mathcal{U}$ such that $A \subseteq U$. ...
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2answers
46 views

How to prove that a subset is discrete?

Please help me to prove that if $F$ is a closed subset of $\mathbb{R}$ such that $x,y\in F$ implies that $x-y\in F$, then $F$ is discrete iff $\alpha=\inf\lbrace x\in F:x>0 \rbrace$>0 and in that ...
2
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2answers
132 views

Clarification regarding basis for a topology

This might be super trivial but I if possible would like some clarification on this topic. I am reading from Munkres' Topology, 2nd edition, page 78 (if interested). My question regards what a basis ...