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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2answers
25 views

Homeomorphism between sets [closed]

Let $D^2 = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$. Is $D^2 \setminus \{p\}$ homeomorphic to $D^2 \setminus \{p, q\}$ such that $p \neq q$? Please explain your answer.
0
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0answers
10 views

Examples of monotone mappings?

I am looking for some interesting (non-trivial) examples of functions between normal spaces which are perfect and monotone, i.e., functions which are surjective and closed preimages of singletons ...
0
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1answer
21 views

Quotient Topology, why is this set “saturated”?

It says $[2,3]$ is saturated with respect to $q$, but not open in $Y$. BUt it doesn't make sense to me because $q(A) = q([0,1) \cup[2,3]) = [0,1) \cup [2-1,3-1] = [0,1) \cup [1,2] = [0,2] = ...
0
votes
1answer
12 views

Question about Lebesgue Covering Dimension

Suppose we have a metric space equipped with two different metrics: $(X,d), (X, d')$. What must be true of the metrics: $d, d'$ in order for $X$ to have the same Lebesgue covering dimension? A ...
1
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1answer
31 views

proof that a set of all bounded real valued functions is complete.

I am trying to understand the proof below. I know that a set A is complete if all Cauchy sequences converges in A. I don't understand 7th line of the proof. Why do we consider particular $x_0 \in X$ ...
2
votes
2answers
74 views

Topology on generalized metric space and metric space

Let $X$ be a nonempty set and $d: X\times X\to R$ be a function such that for all $x,y\in X$ and all distinct $u, v\in X$ each of which is different from $x$ and $y$ (1) $ d(x,y)\geq 0$ ; (2) ...
0
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0answers
27 views

Help with continuum theory

A continuum is a compact connected Hausdorff space (sometimes metric is included in the definition). I have yet to find any references that help me understand composants and components of a ...
2
votes
1answer
42 views

If ${T_n}$ is a sequence of sets that converges to the set of irrational numbers, does $\overline{T_n}$ contain an interval for some $n$?

If $\{T_n\}$ is a sequence of sets that converges to the set of irrational numbers such that $T_1 \subseteq T_2 \subseteq T_3 \subseteq \ldots$. Must $\overline{T_n}$ contain an interval for some $n$? ...
1
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1answer
34 views

$\partial(S') \subset \partial S$ iff $S' \cap S^o \subset (S')^o$

Usually I can come up with some ideas but this time I don't. It would be great if you can tell me how I would make use of the first part of the question to prove the equivalent relation. Question: ...
2
votes
1answer
29 views

Upper and/or lower Bound for Numbers of different topologies on the set $\{1,…n \}$

As the title says I am looking for upper and lower bound for the cardinality of different topologies on a set $\{1,....n\}$ for natural n! Are there some known bounds? My teacher says that there no ...
0
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1answer
24 views

Is the closure of a geodesically convex set convex?

My question is Is the closure of a geodesically convex set convex? If so, is there a simple proof for it? In $ R^n $ there is a simple proof for it through convergent sequences. How should I apply ...
1
vote
1answer
30 views

Topology on the real line 2

"$A\subset \Bbb{R}$ is said convex if for all $x,y\in A$ and $0\leq\lambda\leq1$ then $\lambda x+(1-\lambda)y\in A$. Show that a subset $C$ of $\Bbb{R}$ is convex if, and only if, $C$ is an interval." ...
2
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1answer
61 views

The product topology is the only one on $X \times Y$ that makes the “Maps into Products” theorem valid

The "Maps into Products" theorem says that, (Maps into Products) Let $f: A \to X \times Y$ be given by the equation $$f(a) = (f_1(a), f_2(a)).$$ Then $f$ is continuous iff the functions ...
0
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2answers
27 views

Regarding retractions of $X$ onto subspaces

Let $A \subset X$ be a subspace of $X$. Recall that a retraction of $X$ onto $A$ is a continuous map $r: X \to A$ such that $r(a) = a$ for every $a \in A$. Let $X = \bf R$ endowed with the standard ...
0
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0answers
36 views

Is there other spaces

Is there spaces X which give one-to-one continuous mapping of X onto Housdorff space. I found locally compact space satisfies that, i.e. Every locally compact space X there exist a one-to-one ...
1
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1answer
30 views

Open subset of upper triangular matrices

Let $G$ be the group of all $2\times2$ non singular upper triangular matrices (with matrix multiplication) with entries in $\mathbb{R}$.View G as a topological subspace of $\mathbb{R^4}$.Let U be a ...
1
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2answers
37 views

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected Suppose $G$ is disconnected then $\exists A,B$ relatively open disjoint sets so that $A\neq ...
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0answers
18 views

Topology on the real line [duplicate]

True or false: (Justify) If $A\subset\Bbb{R}$ is open then $A$ is an finite or contable union of open intervals.
0
votes
1answer
24 views

An iff proof concerning complete metric spaces and limits

Prove that $(X,d)$ is a complete metric space, and $B \subseteq X$, then $B$ is a complete metric space iff any sequence $\{a_{n}\} \subseteq B$ that converges in $X$ has a limit in $B$. ...
0
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0answers
34 views

Nowhere density situation [closed]

Let $(X,\tau)$ be a topological space. If $U\in \tau$ and $A$ is a subset of $X$ such that $U\setminus A$ and $A\setminus U$ are nowhere dense, does it necessarily follow that $U\cap A \in \tau$? ...
0
votes
1answer
8 views

Question about irreducible topological spaces

Suppose $X$ is an irreducible topological space, and that $Y$ and $Z$ are closed irreducible subsets of $X$. If $\dim Y=\dim Z$, and $Z \subseteq Y$, does this imply that $Z=Y$?
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0answers
50 views

The space of all continuous mapping.

What is the idea of proving that the space of all continuous mappings of the interval $I=[0,1]$ into the Tychonoff cube $I^\mathbb{R}$ with the compact-open topology is not normal?
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1answer
38 views

Two different topological spaces with the same set of underlying points

This is from an example in Armstrong, "Basic Topology" page 14 #6: Define a subset of the reals to be a neighborhood of a particular real number if it contains that number and its compliment ...
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0answers
13 views

alternative to limit of a mapping between topological spaces.

Let $f$ be a mapping between two topological spaces $X$ and $Y$. $\lim_{x \to x_0} f(x) = y$ is defined as for any open set $U_y$ containing $y$, there exists an open set $V_x$ containing $x$, so that ...
0
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0answers
21 views

Clarification of Regular/Normal spaces discussion in Munkres

In section 31 of Munkres' Topology text on regular and normal spaces, he often assumes for the theorems and interesting results that regular/normal spaces have closed point sets (i.e. the space is ...
1
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3answers
43 views

Equivalent alternative to delta-epsilon formulation of limit?

Is $\lim_{n \to \infty} x_n = L$ same as: $\forall n \in \mathbb N, \exists \ \varepsilon_n > 0 $ so that: $$ \ |x_n - L| \leq \varepsilon_n$$ and $$\varepsilon_n \to 0, \ ...
2
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1answer
28 views

How do I draw this weird space

I'm trying to solve this question: My attempt of solution: Note first I use the stereographic projection to see the sphere is homeomorphic to the plane with a point at infinity, after this we ...
10
votes
1answer
106 views

Is there always an equivalent metric which is not complete?

I have seen that completeness is not a topological property like compactness or connectedness. I have seen some examples also showing that there are two equivalent metrics one of which is complete and ...
1
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0answers
27 views

What are some sufficient conditions for discontinuity of functions of topological spaces?

Let $f: \bf R \to \bf R^\omega$ be a function such that $f(t) = (t,2t,3t,\cdots)$. If $\bf R^\omega$ is endowed with the product topology, it is clear that $f$ is continuous, since every component is ...
0
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1answer
33 views

Closure in a certain topology

Let $\mathcal{T}_3=$ the topology having as basis all open rays $(- \infty, a)$ What is the closure of $A=(2,\sqrt{7})$ in the above topology ?
0
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1answer
34 views

The set of points in complex plane that satisfy a strict linear inequality is open

Let $S = \{(x,y)\in \mathbb C: y > 3x+2\}$. Show that $S$ is an open set. I can imagine what it looks like; a shaded region above a line. I also imagine that we must choose $ε$ (the radius ...
0
votes
1answer
16 views

Denseness of a Preimage

Let $F:\mathbb{R^2} \to \mathbb{R}$ and $A\subset \mathbb{R}$, where $A$ is a dense set on $\mathbb{R}$ ,and $F$ is continuous on $\mathbb{R^2} $. Is $F^{-1}(A)$ a dense set on $\mathbb{R^2}$ ?
0
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1answer
38 views

Tychonoff space is embeddable

How can I prove this theorem: A topological space is a Tychonoff space iff it is embeddable in a compact Housdorff space. Thanks
8
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0answers
91 views

Hatcher Chapter $0$ Exercise $7$

I am going through Hatcher's Algebraic Topology. But I'm stuck with the question $7$ of chapter $0$. I cannot understand the construction of the space $Y$. After the one-point compactification ...
2
votes
2answers
55 views

Show $\mathbb{R} \setminus \{0\}$ is dense and open in $\mathbb{R}$

I was trying to prove the statement that $$\mathbb{R} \setminus \{0\} \text{ is dense and open in $\mathbb{R}$}$$ Could someone help to read my proof and give me some feedbacks/corrections. Here is ...
0
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2answers
36 views

Whether $[0, 1)$ is closed in the lower limit topology

Let T be the smallest topology on $\mathbb R $ which contains all sets of the type $ [a,b)$ where $ a,b \in \mathbb R $ and all sets of type $[a,b)$ are open in T .Determine if the set $[0,1)$ is ...
1
vote
2answers
31 views

Topology - Prove that $[A] = \displaystyle\overline{(X - A)^{\circ}}$

Prove that $[A] = \displaystyle\overline{(X - A)^{\circ}}$ where $[A]$ is the closure of $A$ and $\circ$ is denoted by the interior. Also there is supposed to be a bar over all of $(X - A)$ but it did ...
0
votes
2answers
26 views

The boundary of the union of two sets is a subset of the union of boundaries

I'm stuck on trying to get this proof started. I want to prove that $\delta(S_1 \cup S_2)\subset \delta S_1\cup\delta S_2$, where $S$ is some set. I don't need a full proof, just a hint to get ...
1
vote
1answer
37 views

Example of a convex set whose closure is not convex?

An enumeration $ν\colon ℕ → A$ of the rationals $A$ in $(0..1)$ yields an open set $U_ν = \bigcup_{k ∈ ℕ} B_{1/4^k}(ν(k))$, containing all of $A$. You can choose $ν$ such that $U_ν ⊂ (0..1)$ (by using ...
1
vote
2answers
43 views

Open Ball definition of Closure, Interior, and Boundary

First the definitions in question. Closure: If $A$ is a subset of $\mathbb R ^n$, the closure of $A$ denoted $\bar{A}$, is the set of $x \in \mathbb R ^n$ such that for all $r > 0$ $$B_r (x) \cap ...
3
votes
1answer
56 views

How to draw $S^1\times I$

I always thought $P=S^1\times I$, where $S^1$ is the circle and $I=(0,1)$ with the standard topology is the surface of the cylinder, but I was reading a book which says me another thing: Even, if ...
0
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1answer
45 views

Basis in topology and AC

Let S ⊂ ℘(X). Let T be the coarsest topology on X which contains S. Then we call T the topology generated by S. Let S ⊂ ℘(X). Then we can easily prove using (generalised distributive law) the ...
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0answers
38 views

A question about the interior of a set.

If $(X,d)$ is a metric space and $A \subseteq X$ then the $int(A)$ is the union of all open sets contained in $A$. Then we have that $int(A)$ consists of all interior points of $A$. Everytime I see ...
0
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1answer
20 views

Characterize the following sets as closed/open in the space of M2(R)

Characterize the following sets as closed/open in the space of $M_2(R)$(topologized by considering it as a subset of euclidean space of dimension $4$ in the obvious way ) Set of matrices of the ...
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0answers
31 views

Write the Interior and closure of the following Sets in the space $\{5\}\cup(0,2]$

Write the interior and closure of the following sets in the space $\{5\}\cup(0,2]$ $[0.5,2]$ $(0,0.5] $ $\{5\}\cup(0,0.5]$ I need step by step solution , I missed the class during illness, so ...
4
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1answer
164 views

Two point topological space

Is there a standard name for the two point space with precisely one singleton being the only nontrivial open set? What are its most noteworthy categorical properties?
3
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1answer
52 views

Almost-discrete topological spaces

Call a topological space almost-discrete iff it can be obtained from discrete topological spaces by any combination of: (small) products, (small) coproducts, subspaces, quotient spaces. (Infinite ...
0
votes
1answer
80 views

Showing a function on $\mathbb{R}^2$ is a surjective and continuous

Given an open set $U$ (in the standard topology on $\mathbb{R}^2$) and a function $f:U\rightarrow \mathbb{R}^2$, I would like to show that, assuming $f$ is injective (1 to 1) and continuous: 1) $f$ ...
2
votes
2answers
108 views

Is compactness a generalization of completeness

Is the concept of compact spaces a generalization of completeness to non-metric topological spaces?
0
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0answers
36 views

Textbooks for Algebra, Analysis, Combinatorics, Geometry and Topology :) [closed]

I have the list of some of the areas of Mathematics from wiki: Algebra Analysis Combinatorics Geometry and Topology I want the best textbooks from all of these fields. My compilation: Algebra - ...