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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Prob. 4, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: How to verify the supremum property?

Let $X$ be an ordered set in the order topology. Suppose that $X$ is connected. How to show that $X$ is a linear continuum? My effort: Suppse $x, y \in X$ such that $x < y$. If $(x,y)$ were ...
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1answer
21 views

Union of connected sets

$\forall \beta \in I$, $A_{\beta }$ is connected, and $\left ( \bigcup_{\alpha < \beta }A_{\alpha } \right )\cap A_{\beta }\neq \varnothing$ . Is $\bigcup_{\alpha \in I}A_{\alpha } $connected? For ...
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36 views

GENERAL TOPOLOGY [on hold]

1.Can you give me the counter-example for the theorem "Every compact space is countably compact" . 2.Can you give me the counter-example for the theorem "Every compact subset of a Hausdorff space is ...
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1answer
10 views

Subspace of Lindelöf space is not Lindelöf: Example

The Munkres' topology book provides Example 30.5 (p.193, 2nd Ed) for a subspace of a Lindelöf space that need not be Lindelöf as follows: The ordered square $I_0^2$ is compact; therefore it is ...
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1answer
15 views

Let $f:(X, \mathfrak T_X) \rightarrow (Y, \mathfrak T_Y)$ be a continuous function. Then $f(Cl(A) = Cl(f(A))$.

Let $f:(X, \mathfrak T_X) \rightarrow (Y, \mathfrak T_Y)$ be a continuous function. Then $f(Cl(A) = Cl(f(A))$. My definition of closure is: Let $(X,\mathfrak T)$ be a topological space and let $ A ...
2
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12 views

Informal interpretation of meager sets

I've been wondering if there is a nice informal interpretation of meager sets akin to the respective interpretations I give below to other notions of "small" sets. The general setup to tease out ...
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3answers
575 views

Why can't differentiability be generalized as nicely as continuity?

I was a little bit dissapointed when I learned to differentiate on manifolds. Here's how it went. A younger me was studying metric spaces as a first unit in a topology course, when a shiny new ...
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1answer
12 views

Completely Regular Spaces and Embeddings

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. We were going over separation axioms in class when assigned the following problem. Given ...
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1answer
17 views

Topology induced by a closed-finite topology

Let $(X, \tau)$ be a topological space where $\tau$ is the closed-finite(co-finite) topology. Consider $A \subset X$, is the topology$\tau_{A}$ induced on $A$ by $(X, \tau)$ going to be closed-finite? ...
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1answer
21 views

Let $M\subseteq \mathbb{R}^k$: Manifold topology vs. trace topology

I'm confused about the topology of submanifolds of $\mathbb{R}^n$: Let $M$ be such a $k$-manifold (say, the circle $S^1$, of dimension $1$, embedded in say $\mathbb{R}^7$); the topology of such a ...
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22 views

Stokes Theorem Manifold with Corners Proof

I'm working through the proof for Stokes' Generalized Theorem for Manifolds and have a questions about corners. I've seen several proofs for manifolds with corners by creating diffeomorphisms to ...
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2answers
48 views

Example 5, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: Is this map always continuous?

Let $(X, \Vert \cdot \Vert)$ be a given normed space that has elements other than the zero vector $\theta_X$. And let $T \colon X-\{\theta_X \} \to X$ be defined by $$T(x) \colon= \frac{1}{\Vert x ...
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1answer
21 views

Example 2, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be a linear continuum?

Let $X$ be a well-ordered set, let $[0,1)$ denote the half-open interval (open from the right) on the real line, and let $X \times [0,1)$ have the dictionary order. Then how to show that $X \times ...
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2answers
40 views

Examples about compactness

Compactness implies countably compactness which in turn implies limit-point compactness. Sequentially compactness implies limit point compactness. $Z_{+} \times \{0,1\}$ with two-point indiscrete ...
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15 views

About a map between two topological manifolds with different dimensions

Let $M_1$ be a $n$-dimensional topological manifold and let $M_2$ be a $m$-dimensional topological manifold, such that $m>n$. Moreover, let $U\subset M_1$ be an open set and let $f:U\rightarrow ...
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1answer
21 views

On general topological spaces and $C(X, \mathbb R)$ , where for closed sets $A,B$ in $X$ , $I_A=I_B \implies A=B$

Let $X$ be a metric space and $C(X, \mathbb R)$ be the ring of all real valued continuous functions from $X$ . For $A \subseteq X$ , let us define $I_A :=\{f \in C(X, \mathbb R) : f(x)=0 , \forall x ...
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1answer
28 views

What is the stone space $S_n(T)$ for a theory with infinitely many equivalence classes, each class infinite?

What is the stone space $S_n(T)$ for each $n$, for a theory $T$ with infinitely many equivalence classes, each class infinite.
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1answer
63 views

Brouwer's fixed point continuous function

Can anyone point me out the continuous functions without brouwer fixed point's for the following sets $$A = \{x \in \mathbb{R}^2 | x_1,x_2 \geq 0 \text{ and }x_1^2+x_2^2 = 1 \}$$ $$B = \{x \in ...
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1answer
14 views

Bijectivity of radial projection

So I'm trying to show that the boundary of a simplex is homeomorphic to a sphere, and I want to do it by radial projection. But it's turning out to be surprisingly difficult. Intuitively, it is clear ...
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2answers
48 views

How can $0$ be an interior point of $[0,1]$ when $\mathbb R$ is given the discrete topology?

Let $\mathbb R$ be topologized with the discrete topology. Then every subset of $\mathbb R$ is clopen. So, for every $A \subset \mathbb R$, $\operatorname{int}(A)=A$. But if $A=[0,1]$, the ...
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2answers
41 views

Confusion over the concept of “compactness”

I have to prove some stuff that involves the concept of collection, in particular those relating to compact sets. But then I have got this trouble. For example, consider the set of all rationals. If ...
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23 views

Prob. $ 9 $, Sec. $ 23 $ of Munkres’ “Topology”, $ 2^{\text{nd}} $ Ed.: How to show this subspace is connected?

Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, how to show that $$\left(X \times Y \right) \setminus \left(A \times B \right)$$ is also ...
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1answer
26 views

The set of points of continuity of a real-valued function on a metric space is a $G_\delta$ set

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. I know lots of other ...
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2answers
43 views

The distance between two sets does not change if closure is taken

Given $ (X, d)$ a metric space, $ A, B \subset X$, show that $ d(A, B)=d (\overline {A}, B) $. I'm not being able to show that $ d(A,B) \leq d (\overline {A}, B) $. Can anybody help me? The set ...
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0answers
21 views

Basis for a topology of a scheme

Suppose that $X$ is a proper and connected scheme over an algebraically closed field. Moreover let $\mathcal A$ be a collection of open subsets of $X$ with the following property: For every open ...
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1answer
44 views

Some question about path connectedness

I think that's intuitively evident but I can't prove that the set $\mathbb{S}^n\setminus\{(0,\cdots, 1),(0,\cdots, -1)\}\; (n>1)$ is path connected. Does anyone have a formal argument to prove it? ...
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1answer
31 views

Regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $.

I tried to draw the regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $. I think the regular covering space is: Is it true? How do you draw the non-regular ...
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1answer
25 views

Prove that a function is open

Let $X,Y$ metric spaces and $U \subset X , V\subset Y$ open sets. Let $f:U\rightarrow V$ be a homeomorphism. Prove that $f$ is an open map. I need to show that for every open subset of $U′⊂U$, $f(U′)$ ...
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1answer
37 views

Let $A = \{1- \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$.

Let $A = \{1 - \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$. I am supposed to figure out if this set is closed under certain topologies. I know that means I ...
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2answers
14 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
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18 views

Operator norm on the space on linear functions between Euclidean spaces.

*I'm reading a text which has a preliminary section on Linear maps. I have come across a conclusion that I can't seem prove by myself. * Let $Lin(\mathbb{R}^m,\mathbb{R}^n)$ be the space of linear ...
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3answers
206 views

Confusion about concept of basis in point set topology.

I'm afraid that I have a big misunderstanding about the notion of basis in general topology. For a given topology $\tau$ of set $X$, if there is a collection $S \subset \tau$ of open subsets of $X$ ...
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118 views

Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
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Probs. 2 (d) and 2(e) in Supplementary Exercises, Chap. 2 in Munkres' TOPOLOGY, 2nd ed: How are these maps continuous?

Let $S^1$ denote the set of all complex numbers $z$ such that $\vert z \vert = 1$ (regarded as a subspace of the complex plane), and let the map $f \colon S^1 \times S^1 \to S^1$ be defined by $$f(w ...
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2answers
170 views

Do two closed subsets of $[0, 1]$ with measure $\frac{1}{2}$ intersect?

Let $A$ and $B$ be two closed subsets of $[0,1]$, each with a length of $1/2$. Is it always true that $A\cap B\neq \emptyset$? My intuition is yes, because: Either they intersect in their interior; ...
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1answer
17 views

Continuity of product maps

Let $J$ be a given (countably or uncountably infinite) index set. Let $\{\ X_\alpha \ \colon \ \alpha \in J \ \}$ and $\{\ Y_\alpha \ \colon \ \alpha \in J \ \}$ be collections of topological ...
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0answers
27 views

Show if orbit is discrete, the orbit is closed.

Given $\beta <$ the isometry group $\mathbb{R}^2$. Show that if an orbit $\beta_x$ is discrete, then $\beta_x$ is closed. I am just looking for some feedback and critique of my attempt at a ...
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1answer
23 views

Is there any simply connected polyhedron with a not simply connected face?

According to Wikipedia, For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2. Is it really necessary to specify here, that ...
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1answer
47 views

Comparing different topologies on the Hilbert cube $H = \prod_{n \in \mathbb{N}} [0,\frac 1n]$

This is essentially exercise 8(c) from section 20 (p.128) of Munkres's Topology: Let $X$ be the set of all the sequences $(x_n)$ of real numbers such that the series $\sum_{n=1}^\infty x_n^2$ ...
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48 views

Is every point in the set defining some curve or line in $\mathbb{C}$ or $\mathbb{R}$ a boundary point?

Might be a dumb question but is every point in the set defining some curve or line in $\mathbb{C}$ or $\mathbb{R}$ a boundary point? I reason that it should be since any point in the set has to be ...
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2answers
40 views

Construct an open cover of S with no finite subcover

Let S be a subset of Rn, and suppose that S is not bounded. Construct an open cover of S with no finite subcover, then prove this claim about your open cover. Let S be a subset of Rn such that S is ...
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1answer
31 views

Prove that $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$ is compact and connected

Let be $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$. I have proved that $X$ is a submanifold of $\mathbb{R}^4$ of dimension $3$. I have to prove that $X$ is compact and connected. My idea, thinking of ...
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3answers
47 views

how do you prove the set of accumulation points of Q is R.

I know that the set of accumulation points for the rational numbers is the real numbers, but I'm not sure how to prove this. I need to use the definition: $x$ is an accumulation point of $S$ if, for ...
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1answer
21 views

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$.

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$. My definition of closure is "Let $(X, \mathfrak T)$ be ...
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27 views

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find $\overline A$, int$(A)$, and bdry$(A)$.

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find closure of $A$ $(\overline A)$, interior of $A$ (int$(A)$), and boundary of $A$ (bdry$(A)$). $A$ ...
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1answer
13 views

Let $(X, \mathfrak T)$ be topological space and suppose that A and B are subsets of X such that $A \subsetneq B$. Then $Int(A) \subsetneq Int(B)$.

Let $(X, \mathfrak T)$ be topological space and suppose that A and B are subsets of X such that $A \subsetneq B$. Then $Int(A) \subsetneq Int(B)$. ( $\subsetneq$ means "is a proper subset") My ...
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1answer
37 views

Fractal dimension of a dense subset

Let $M$ be a metric space and $S\subset M$ a dense subset. For vague reasons (below), it seems to me that the upper box-counting dimension of $S$ should be equal to that of $M$, but I don't quite see ...
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0answers
28 views

Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
0
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1answer
45 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then A =∅ or $A = X$ .

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then $A = \emptyset$ or $A = X$. I am studying introduction to proofs and we have learned ...
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0answers
42 views

Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I ...