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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Proof of Supporting Hyperplane Theorem from basic definitions.

My purposes in posting this question are twofold. First, I would like to have a lemma which I have proven on the way to proving the Supporting Hyperplane Theorem checked for rigor (zero tolerance for ...
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1answer
79 views

Aronszajn lines

Exercise 32 of chapter 2 of Kunen (1980) tells me to show that there exists a total ordering with no $\omega_1$ strictly increasing/decreasing sequencies such that every separable subspace is nowhere ...
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3answers
74 views

Which of $(-\infty,\infty]$ and $[-\infty,\infty]$ is homeomorphic to $S^1$?

Is it correct to say that $(-\infty,\infty]$ is homeomorphic to $S^1$? or it is $[-\infty,\infty]$? (considering standard topology). Would you please provide some explanation or better a rigorous ...
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2answers
67 views

What is the difference(s) between $(a,\infty)$ and $(a,\infty]$?

I am studying H. L. Royden's Real Analysis which includes some introduction to Measure Theory; and I encountered $(a,\infty]$ instead of $(a,\infty)$ for the first time! What is the difference(s) ...
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72 views

In an Euclidean $\mathbb R^n$ space, is every compact set an open set? Is it possible to have sets that are both open and bounded?

I know that compact sets are the ones that are both bounded and closed (Heine-Borel Theorem), but since closed and open are not opposites, I cant see if and how a compact set, or a bounded set, can be ...
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32 views

Topological spaces with the same underlying set and basis. [closed]

Let $(X; T_1)$ and $(X; T_2)$ be topological spaces with the same underlying set. Let $B_1$ and $B_2$ be bases for $T_1$ and $T_2$ respectively. Then $T_1$ = $T_2$ if and only if $B_2\subseteq ...
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3answers
147 views

Show the following set is connected

For any $x \in \Bbb R^n$ how do I show that the set $B_x := \{{kx\mid k \in \Bbb R}$} is connected. It should also be concluded that $\Bbb R^n$ is connected. I was thinking of starting by assuming ...
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29 views

Is $\tau$ a topology on $\mathbb{R}^2$? where the elements of $\tau$ are $\emptyset$ and the complements of finite sets of lines and points [closed]

Prove that ($\mathbb{R}^2$,$\tau$) is a topological space where the elements of $\tau$ are $\emptyset$ and the complements of finite sets of lines and points I don't know how to prove the second ...
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19 views

Find the boundary and interior of subsets of $\mathbb{R}^2$ [closed]

Find the boundary and interior for each of the following subsets of $\mathbb{R}^2$: -$A =\{ (x,y) \in \mathbb{R}^2 \colon y = 0 \}$, -$B = \{ (x,y) \in \mathbb{R}^2 \colon x > 0, y \not=0 \}$.
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41 views

Properties of closure and example

Let $A_1, A_2, A_3, \dots$ be subsets of a metric space. If $B=\bigcup_{i=1}^\infty A_i$, prove that $\overline{B}\supset \bigcup_{i=1}^\infty \overline{A_i}.$ Show, by an example, that this ...
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16 views

Construct an example in which $x$ is $\tau_1$-accumulation point of a subset $A$ of $X$ but It is not $\tau_2$-accumulation point of $A$

Let $\tau_1$ and $\tau_2$ be a topologies on a set $X$ with $\tau_1 \subset \tau_2$ Construct an example in which $x$ is $\tau_1$-accumulation point of a subset $A$ of $X$ but It is not ...
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53 views

Proof $\{(x,y,z)|4x^2+9y^2+16z^2<1\}$ is an open set

In order to prove that the points $(x,y,z)$ such that $$4x^2+9y^2+16z^2<1$$ form an open set, I tried this: Pick a generic point of the ellipsoid, lets say $$4x^2+9y^2+16z^2$$ Now, I'll form ...
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3answers
210 views

Is a ball noncompact?

A compact manifold usually refers to "a manifold without a boundary", for example the usual 2-sphere $S^2$. What about a manifold with a boundary? Intuitively, I think such an example, e.g. a ball ...
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1answer
33 views

Topology (Basis)

A basis for a topology is defined as the subcollection of the topology such that every member of the topology can be expressed as the union of members of that subcollection. But if the basis doesn't ...
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0answers
19 views

Is the set of limitpoints of a set always closed? [duplicate]

If $E$ is a subset of a topological space and $E'$ denotes its set of limitpoints then I can prove that $E'$ is closed under the extra condition that the topological space is $T_1$. For a proof see ...
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1answer
33 views

About expansive homeomorphim

We say $(X,f)$ is expansive if there is $c(f)>0$ such that if $d(f^{n}(x), f^{n}(y))< c(f)$ for every $n\in Z$ then $y=x$. Let $(X,f)$ is expansive with constant $c(f)$ and for infinite set ...
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1answer
16 views

What 'limit point' means in $\Bbb Z_+\times {\{a,b}\}$?

Here was a question, but raised another question of meaning of a neighborhood. According to the answer: [Let] $Y=\{a,b\}$. If $S$ is a subset of $\Bbb Z_+\times Y$ and $(n,a)\in S$, then $(n,b)$ ...
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34 views

The set of all limit point of a set

Let $E'$ be the set of all limit points of a set $E$. Do $E$ and $E'$ always have the same limit points? Proof: $E''\subset E'$ because $E'$ is closed. But inclusion $E'\subset E''$ is false. We ...
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1answer
73 views

Proof that $f$ is an isometry

Let $D$ be the set of points in $\Bbb R^2$ such $\lvert p \rvert\leq1$, and let $f: D\rightarrow D$ be a surjective function, satisfying this relation: $\lvert f(p)-f(q)\rvert \leq \lvert p-q\rvert$, ...
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99 views

A Question Regards Quotient Topology and Vector Space

Let $V$ be a $n$ dimensional vector space,Consider the topological space $(V, \mathcal{T}_v)$, where $\mathcal{T}_v$ is standard topology on $V$. Standard topology on $V$ is defined by ...
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34 views

homeomorphism as a result of other homeomorphisms

If $$B = \bigcup_{R>0} B_R$$ and all the identities $$\operatorname{id}_R : (B_R,d_1) \rightarrow (B_R,d_2)$$ for $R>0$ are homeomorphisms, then is $$ \operatorname{id} : (B,d_1) \rightarrow ...
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2answers
66 views

Can we define $ℝ^A$ where A is uncountable?

The question is pretty straightforward. How can we define the expression $ℝ^A$ when $A$ is an uncountable set? For example what is defined by forms such as $ℝ^ℝ$ or $ℝ^ℂ$? If $A$ is countable,then ...
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2answers
71 views

Is a continuous function $f : \mathbb{Q}\to\mathbb{Q}$ always bounded on a closed interval?

Can a function $f : \mathbb{Q} \to \mathbb{Q}$ that is continuous on an interval $[a,b]$ not be bounded on $[a,b]$? I'm asking this because in Spivak's Calculus, the "Boundedness Theorem", which ...
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1answer
76 views

Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?

So it is weekend! and I am reading a nice book, "The Poincaré conjecture", written by a mathematician (Donal O'Shea, topologist). The book introduces step by step basic concepts of Topology, and talks ...
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1answer
22 views

How is the general concept of a basis useful?[topology] [closed]

In linear algebra a basis on a finite vector space helps yield a richer description of that vector space [allowing for the concept of dimensions for example]. In what ways does defining a basis for ...
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2answers
51 views

Properties of a set of all limit points

Let $S'$ be the set of all limit points of a set $S$. Prove that $(A\cup B)'=A'\cup B'$. Proof: Let $x\in (A\cup B)'$ then $x$ is a limit point of $A\cup B$. Then any deleted neighborhood $N'(x)\cap ...
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20 views

show that the ordered square is locally connected~

show that the ordered square is locally connected but not locally path connected. what are the path components of this space? this problem is exercise munkres 25-3, and also example 24-6 and ...
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30 views

Real Projective Space Homeomorphism to Quotient of Sphere (Proof)

I need to construct a function $f : (\mathbb{R}^{n+1}-\{0\})/{\sim} \to S^n/{\sim}$, by $$f ([x]_{\mathbb{RP}^n}) = \left[\frac{x}{\|x\|}\right]_{S^n/{\sim}},$$ where $S^n = \{ x \in ...
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1answer
38 views

Intuition on the Topological definition of continuity, considering the special case of the step function.

I'm trying to get an intuition for open sets and topological reasoning in general. One example I want to understand is the step function, and specifically why it would be considered discontinuous ...
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28 views

The set of limit points is closed

Let $E'$ be the set of all limit points of a set $E$. Prove that $E'$ is closed. Let $z$ be a limit point of $E'$. Then for any $\varepsilon>0$ deleted neighborhood with radius $\varepsilon/2$ has ...
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Denseness: Closed Space

I need this as lemma. Topological Space Given a topological space $\Omega$. Consider a closed space: $$\mathcal{S}\subseteq\Omega:\quad\mathcal{S}=\overline{\mathcal{S}}$$ Then for dense domains: ...
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37 views

removing rationals in topolosist's sine curve

Munkres pg. 160 says, where $S$ is the topologist's sine curve If one forms a space from $\bar S$ by deleting all points of $V$ having rational second coordinate, one obtains a space that has only ...
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31 views

Weakening compactness in metric spaces

Is the following true in a general metric space $X$? Every net (in $X$) of cardinality $\kappa$ contains a convergent subnet of cardinality $\kappa$ if and only if every open cover of $X$ admits a ...
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3answers
363 views

Is the complement of a closed set always open? [duplicate]

My book says that that a set is closed if its complement is open. Can a set be closed for other reason or is this if supposed to be an iff? Is there a way to prove this statement?
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1answer
37 views

Counterexamples of Brouwer fixed point theorem applied on the close unit ball

Brouwer fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least one fixed point. Brouwer fixed point theorem applies in particular on the ...
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33 views

Other counterexamples for this problem on connectedness

If $A$ is a connected subspace of $X$, does it follow that $\operatorname{Int}A$ and $\operatorname{Bd}A$ are connected? Does the converse hold? Justify your answers. For original problem, I can ...
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1answer
32 views

Proof on Multiplication Funtion's Continuity.

Suppose $(X, \mathcal{T}_X)$ is a topological space, and that $f_1 : X \to \mathbb{R}, f_2: X \to \mathbb{R}$ are continuous functions. prove the multiplication function $$f_1f_2 : X \to ...
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3answers
33 views

Discrete metric means all sets are countable?

I was working on a proof of "Show that if $A \subseteq \Re^2$ is discrete, then A is a countable set." and I thought about using the discrete metric ($d(x,y)=\delta_{xy}$) on the set as an example ...
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1answer
85 views

The boundary of a closed subset $A$ of a compact connected hausdorff space

Let $X$ be a compact connected hausdorff space. Let $A\subseteq X$ such that $A$ is closed and $A\neq\emptyset,X$. Let $C$ be a (connected) component of $A$. Is it true that $C\cap\partial ...
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50 views

A bijective continuous map is a homeomorphism iff it is open, or equivalently, iff it is closed. [closed]

Wikipedia states that "a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.". How do we prove this fact? I can prove the obvious ...
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14 views

If $S$ and $T$ are closed vector subspaces then $S+T$ is closed [duplicate]

Let $V$ be a Banach normed space, $S,T \subset V$ be closed vector subspaces. Assume $\operatorname{dim}(T)<\infty$. Show that $S+T$ is closed. So I encountered this problem trying to use ...
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39 views

a theory of transcendental functions?

Lately I've been interested in transcendental functions but as I tried to search for books or articles on the theory of transcendental functions, I only obtained irrelevant results (like calculus ...
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1answer
21 views

Re)order topology and subspace topology…

let $X=(-\infty, -1) \cup [0,\infty)$, subspace of $\mathbb{R}$. Then is it different from the order topology? Say $(-1/2,1) \cap X =[0,1)$ is open in $X$, but not open in the order topology?? I ...
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26 views

order topology and subspace topology

let $X=(-\infty, -1) \cup [0,\infty)$, subspace of $\mathbb{R}$. Then is it different from the order topology? Say $(-1/2,1) \cap X =[0,1)$ is open in $X$, but not open in the order topology??
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1answer
25 views

Show, by an example, that $h$ need not be continuous

Let $X$ and $Y$ be a topological space with $X=E \cup F$ Let $f: E \to Y$ and $g:F \to Y$ with $f=g$ on $E \cap F$ be continuous respect to the relative toplogies. Note that $h=f \cup g$ is a function ...
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1answer
38 views

How does this imply “the unit balls completely determine the metric space structure”?

Let $X$ be a metric space. Suppose you know that $$B[x,r] = x + rB[0,1]\qquad\text{and}\qquad B(x,r) = x + rB(0,1)$$ where $B[x,r]$ and $B(x,r)$ are respectively the closed and open balls in $X$. How ...
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1answer
117 views

Existence of a bounded ball

Lets define: $$F(X) :=\{A \subseteq X \mid A \neq \emptyset , A = \overline{A}\}.$$ For $A, B \in F(X)$ and $p \in X$ define $$d_p(A,B) = \sup_{x \in X} \{ | \operatorname{dist}(x,A) - ...
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21 views

Show: if $x \in A'$, then $x \in (A \setminus \{x\})'$ in Hausdorff space; dash denotes limit points

EDIT: It was pointed out I managed to misstate the question. Sorry, and thanks to Daniel Fischer for pointing it out. RTP: Letting $B'$ be the set of limit points of $B$ for any set $B$, show that ...
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2answers
62 views

can you show example for this?

Show that if $X$ is a well-ordered set, then $A=X \times [0,1)$ in the dictionary order is a linear continuum solution link https://onesidey.wordpress.com/2012/03/09/munkres-24/ and I wonder ...
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1answer
47 views

Prove that the image of a a closed and bounded interval in $\mathbb{R}$ is a a closed and bounded interval in $\mathbb{R}$?

According to the excercise 7.22 of the book Topology by Franzosa: Combining the Extreme Value Theorem and the Intermediate Value Theorem, prove the following theorem: Let $[a, b]$ be a ...