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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1answer
23 views

Is there a name for a property defined in terms of open sets?

We know that if a property is defined in terms of open sets then the property is preserved under a homeomorphism. Is there a name for such a property?
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1answer
11 views

A Hausdorff space which is not completely regular

My example is, $f : \mathbb{R}^+ \to \mathbb{R}$ defined by: $$f(x) = \begin{cases} x, &\text{if }0 \leq x < 1 \\ \tfrac{1}{x}, &\text{if }x \geq 1. \end{cases}$$ Even though $f(0)=0$ but ...
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3answers
33 views

Show that $S=\mathbb R^2\setminus\{(a,b):a,b\in\mathbb Q\}$ is path connected. [duplicate]

Show that $S=\mathbb R^2\setminus\{(a,b):a,b\in\mathbb Q\}$ is path connected. By definition of path connected, there should exist continuous mapping $f:[0,1]\rightarrow \mathbb R^2$ s.t. ...
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2answers
22 views

Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
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1answer
41 views

Prove that a set is compact

Let $X$ be a compact space, let $U$ be an open set in $X$, Let $f:U\to [0,1]$ be a continuous map. Prove that the set $$K=\{(x,t): x \in U , 0 \leq t \leq f(x) \} \subset X \times [0,1]$$ is compact. ...
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0answers
37 views

Proving a set is dense in R if all limit points are in R

Prove: $E$ is dense in $\Bbb R$ if and only if the set of limit points of $E$ equals $\Bbb R$. That is, $$E′=\{x\mid \hbox{$x$ is a limit point of $E$}\}=\Bbb R\ .$$
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1answer
25 views

True or False: Topological Group and $S^1 \vee S^1$

$i.$ $S^1 \vee S^1$ can be embedded in a topological group $ii.$ $S^1 \vee S^1$ can be covered by a topological group I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which ...
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0answers
37 views

Prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$

I need to prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$. I define the map $h:\mathbb{R} \times S^1 \to \mathbb{R^2} \setminus \{(0,0)\}$ by ...
4
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3answers
40 views

Relation between continuous maps and convergence of sequences

I am studying metric spaces and I know that in a normed space $E$ a map $T:E \to E$ is contínuous if and only if $T(x_n) \to T(x)$ for every convergent sequence $x_n \to x$ in $E$. In my notes there ...
0
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1answer
42 views

Is there a discrete initial topology on the set of real numbers?

Consider the real numbers R first as just a set with no structure. Then consider it as a topological space R* with the usual topology. The question is: is there a function f from R to R* whose ...
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0answers
27 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
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3answers
37 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
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2answers
55 views

On invertible matrix in $\mathbb R^{n^2}$ [on hold]

How do i prove that the invertible matrix form an open and disconnected set in $\mathbb R^{n^2}$ or generally if $G$ its a multiplicative group of matrices in $\mathbb R^{n^2}$ with Int($G$) non ...
1
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1answer
16 views

Constructing semi-regular spaces

This is essentially a question from Engelking's text. Suppose that $( X , \mathcal{O} )$ is a (Hausdorff) space, and let $\mathcal{O}^\prime$ be the topology on $X$ generated by the family of all ...
5
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1answer
50 views

Show that there is sequence of homeomorphism polynomials on [0,1] that coverge uniformly to homeomorphism function.

Let $f:[0,1]\rightarrow [0,1]$ be a homeomorphism. Show that , there exists a sequence of polynomials $$(P_n(x))_n$$ such that $P_n(x)$ converge uniformly to $f$ on $[0,1]$ and every $P_n(x)$ is a ...
0
votes
1answer
17 views

Show that T2-space is preserved by continuous map. [duplicate]

Let (X,$\tau$) and (Y,$\tau_1$) be topological spaces and f : (X,$\tau$)$\rightarrow$(Y,$\tau_1$) a continuous map. If f is one-to-one, prove that (Y,$\tau_1$) is Hausdorff implies that ...
0
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1answer
78 views

All topologies on $X=\{ a,b \}$ [duplicate]

I am trying to find the possible topologies on $X=\{ a,b \}$. $\varnothing ,\{ a,b \}$ $\varnothing ,\{ a \},\{ a,b \}$ $\varnothing ,\{ b \},\{ a,b \}$ $\varnothing ,\{ a \},\{ b \},\{ a,b \}$ ...
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1answer
48 views

Correct proof of supremum property?

Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$. Prove that $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$. ...
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0answers
22 views

Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
4
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0answers
37 views

Gluing two solid tori along their boundary resulting in a topological manifold

The following question is from a past qualifying exam. Take two solid tori $D^2 \times S^1$, and construct the space $X$ by identifying their boundaries via the map $f \colon \partial D^2 \times S^1 ...
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1answer
56 views

Lebesgue covering dimension of $[0,1]$

Say, we define the Lebesgue covering dimension (LCD) like this: A set $S\in \mathbb R^n$ has LCD $d\in \mathbb N$ if and only if $d$ is the smallest natural number such that for any open cover ...
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0answers
21 views

Family of Morse functions made constant

I'm looking for a proof of the following theorem: Let $f_t$ be a family of real-valued Morse functions defined on a smooth compact manifold $M$, and where $t$ is in $[0,1]$ (So for all value of $t$, ...
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0answers
30 views

The set of rational numbers, each point is point accumulation

Please let us help someone by telling you a precise formulation is below, and then someone please tell me solution that has since become like that with a few days my friend we debates, here my ...
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0answers
31 views

A topology defined on collections of open covers of a topological $X$.

Is anyone familiar with a topology which is defined on collections of open covers of a topological space $(X,O)$? I am trying to define a topology induced by a linear ordering of the open covers, ...
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1answer
77 views

Can an n dimensional object cover an n+1 dimensional object?

Is it possible for an n dimensional object to ever cover an n+1 dimensional object? For example, could a square ever cover a cube? Note: Definition of "cover" here means to completely cover the ...
0
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0answers
19 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
0
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1answer
51 views

A question about the proof of an obvious result

This is obviously true that a local homeomorphism is a continuous map. I tried to prove it this way : Suppose $f:X \to Y$ is a local homeomorphism, then $f$ is continuous if for each $x\in X$ and ...
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5answers
72 views

If $A$ and $B$ are compact subset of $\mathbb R$ , then so is $A+B$.

Prove the following: If $A$ and $B$ are compact subset on $\mathbb R$ , then so is $A+B:= \{a+b\mid a\in A ,b\in B\}$. I was actually thinking about first proving that if $A\subseteq \mathbb R$ is ...
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0answers
31 views

Injective and continuous function that is an embedding

Consider $n,d\in \mathbb N$ and $N= {n+d\choose d}-1$, then the well known $d$-uple embedding: $$\rho_d: \mathbb P^n(\mathbb C)\longrightarrow\mathbb P^N(\mathbb C)$$ is a continuous (respect to ...
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1answer
26 views

Show that a map f : (X,$\tau$) $\rightarrow$ (Y,$\tau_1$) is continuous if and only if $f^{-1}(U)\in\tau$ , for every $U\in$B1

Let (X,$\tau$) and (Y,$\tau_1$) be topological spaces and B1 a basis for the topology $\tau_ 1$. Show that a map f : (X,$\tau$) $\rightarrow$ (Y,$\tau_1$) is continuous if and only if ...
0
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1answer
89 views

Question about proof on basis

I found this proof online, but I have a bit of trouble understanding it. Question: Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an ...
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1answer
62 views

Topology question about open spaces of a topological space homeomorphic to the full set. [on hold]

Let $\mathcal{U}$ be an open subset of $\mathbb{R}^m$ such that there is homeomorphic $f$ from $\mathcal{U}$ to $\mathbb{R}^m$ and also $f$ is an uniformly continous function. Show that ...
0
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2answers
64 views

Construction of an embedding of $\mathbb{Z} \cup \{\infty\}$ into $\mathbb{R}$.

Let $X$ be the one-point compactification of the integers $\mathbb{Z}$, construct an embedding of $X$ into the reals $\mathbb{R}$. I already appreciate your hints/answers. Thanks
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0answers
41 views

Proof that a correspondence is upper hemicontinuous if and only if it's graph is closed

I'm working through a textbook (General Equilibrium Theory) where proofing the following theorem is left as an exercise to the student - unfortunately I dont know how. Theorem 23.1: (A ...
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0answers
63 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
3
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2answers
70 views

Riemannian manifolds are metrizable?

I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. ...
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3answers
286 views

Given an example of a metric space in which every sphere has two centers

This is a question from Wilansky "Topology for analysis", P.15 Prob. 103 Maybe I was thinking too Euclidean, I can't come up some other "centers" of the sphere :(
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2answers
34 views

How Construct Clopen-Compact Bitopological Spaces?

Dear all who love general topology, In general topology we know the notion of clopen-compact spaces (introduced by A. Sostak): a topological space $(X,\tau)$ is called clopen-compact if every clopen ...
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2answers
32 views

an example of a continuous bijection which is not a homeomorphism [duplicate]

I need an example of a continuous bijection $f:X \to Y$, where $X$ is NOT compact and $Y$ is Hausdorff, such that $f$ is not a homeomorphism. (It is easy to show that if $X$ is compact, then $f$ is ...
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1answer
14 views

About the interior ball condition of a convex set with C^1 boundary

Let $\Omega$ an open bounded and convex domain in $R^n$. Suppose that the boundary of this set is $C^1$. Then $\Omega$ satisfies the interior ball condition for all boundary points? Intuitively ...
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4answers
162 views

The set of points where two continuous functions agree is closed.

I want to prove that if $f,g$ are continuous functions from a topological space $(X,\tau)$ to a metric space $(Y,d)$ then the set $$ A = \{ x \in X : f(x) = g(x) \} $$ is closed. I found a very ...
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2answers
47 views

Closed Intervals

How do topologists prove continuity of a function with the usual topology at the endpoints of a closed interval? For instance, how would a topologist prove continuity for $f(x)=x^2$ on the closed ...
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1answer
35 views

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$.

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$. I need to verify correctness of my proof and ask if there is a more straight-forward ...
0
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1answer
24 views

Fundamental group smash product

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is? probably there should be a general way like there is for the wedge sum due to van ...
4
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0answers
32 views

Properties of first-countable spaces

Hi I have questions regarding first-countable spaces. I just want to confirm something: The following are properties regarding limits and continuity of first countable spaces on Wikipedia: If $f$ ...
4
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1answer
66 views

Fundamental group of quotient of $S^1 \times [0,1]$

I have a past qual question here: Let $X = S^1 \times [0,1] /{\sim}$, where $(z,0) \sim (z^4,1)$ for $z \in S^1 = \{ z \in \mathbb{C} \colon \| z \| = 1 \}$. Compute $\pi_1(X)$. I've been trying to ...
0
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1answer
44 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
2
votes
2answers
61 views

Show that two spaces are not homeomorphic

Let $H=[-1,1]\times \{0\}$ and $V=\{0\}\times [-1,0)$ in the plane. Let $T=H \cup V$. Show that $T$ is not homeomorphic to the unit interval $I=[0,1]$. My idea for this problem is that , if we remove ...
2
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2answers
34 views

Topology and Arithmetic Progressions

I'm self-studying from "Elementary Topology Problem Textbook" by O.Ya.Viro et al. This is Exercise 2.Lx : Consider the following property of a subset $F$ of the set $\mathbb{N}$ of positive ...
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1answer
28 views

Show that composition of continuous function is continuous in product topology.

Suppose $H: X \times I \to Y$ is a continuous map of topological spaces $X,Y$ and $I = [0,1]$. And suppose $K: Y \times I \to Z$ is also a continuous map of topological spaces. I want to show that ...