1
vote
1answer
37 views

Intuition behind homeomorphism from $B((0, 0), 1) \to \mathbb{R^2}$

In my notes I have that the following function is a homemorphism from $B((0, 0), 1) \to \mathbb{R^2}$ $$h(x, y) \to \frac{f(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} (x, y)$$ where $f = ...
2
votes
1answer
27 views

Continous surjective map from $S^1$ to $S^n$

Is there any continous surjective map from $S^1$ or $[0,1]$ onto $S^n$, for some $n\geq 2$. Thank you.
0
votes
1answer
41 views

Weakest topology equivalence

Prove the equivalence of the following $Y \subset X$ has the subspace topology . $Y$ has the weakest topology to make the inclusion $i:Y\to X$ continuous. For all topological spaces $Z$ and maps ...
3
votes
0answers
24 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
1
vote
2answers
50 views

sequential continuity vs. continuity

A short and hopefully simple question for someone with more experience in topology: If a topology is induced by a mode of convergence and in fact nothing more is known apriori, wether if this topology ...
2
votes
1answer
39 views

Equivalence of continuous functions

Consider two topological spaces, $X$ and $Y$, and two continuous functions $f$ and $g$. By definition, given an open set $S$ in $Y$, the pre-image of $S$ under $f$ (or $g$) is an open set of $X$. Let ...
1
vote
2answers
81 views

on Continuous and Open Functions

Let $X,Y$ be compact Hausdorff spaces. If $f$ is a continuous function from $X$ onto $Y$, then $f$ is open. I am asking can the above result be proved. I am aware of the following cases: If $f$ ...
0
votes
1answer
67 views

Uniform Continuity $\implies$ Continuity

In metric spaces it is a well known fact that uniformly continuous functions are indeed continuous at any point. What about uniform spaces? How can I prove this? (with the definition of topology in ...
6
votes
1answer
77 views

Topology/continuous functions of $\mathbb{R}^n$ using paths.

We know that for a function $f:\mathbb{R}^n\to\mathbb{R}$ to be continuous, it is not sufficient for itto be continuous with respect to each coordinate. I believethe most commom counter-example is the ...
2
votes
1answer
54 views

Proving the metric attains a minimum on a compact subset

Let $(X,d)$ be a complete metric space. Suppose $B \subset X$ is compact. Prove that for every $a\in X$ the minimum $\min_{b\in B} d(a,b)$ exists. I'm pretty sure you can do this by just using the ...
1
vote
2answers
38 views

Is $f:\mathbb E^1\to X$ continuous?

$f(x)=x$. $X$ is the set of all real numbers with finite complement topology (A set is open in this space iff it's complement is finite).
3
votes
1answer
27 views

Show that $Y$ is not path-connected

Let $\mathbb{R}^2$ with the usual topology and let $$ Y = A_0 \cup (\bigcup_{n \in \mathbb{N}} A_n) \cup (\bigcup_{n \in \mathbb{N}}L_n)$$ where $$ A_0 = \{ 0 \} \times [0,1] \qquad A_n = \{ ...
5
votes
1answer
214 views

Topology: Opens vs Neighborhoods

Disclaimer: This thread is meant informative and therefore written in Q&A style. The problems are highlighted in bold face. The axiomatization of topology can be done in various ways all of ...
0
votes
1answer
24 views

Tomae's Popcorn Function: Preimage of Opens?

I'm just wondering what the preimage of an (open) neighborhood say $(-0.5,0.5)$ containing the point $T(\frac{1}{\sqrt{2}})=0$ under Tomae's popcorn function $T$ looks like. Does somebody have an ...
0
votes
1answer
18 views

semicontinuity implies sequential semicontinuity

I have that $F:X\to (-\infty,+\infty]$, with $X$ topological space. By definition, $F$ is lower semicontinuous in $x_0 \in X$ if $\forall t \in \mathbb{R}: \. t<F(x_0) \.\exists U\in ...
1
vote
2answers
73 views

Continuous Map: Open $\iff$ Closed? [closed]

Is it true that a continuous map is open iff it is closed: $$f\text{ continuous}:\quad f\text{ open}\iff f\text{ closed}$$ The idea is that when somebody asks for embeddings, quotient maps and ...
0
votes
3answers
56 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
0
votes
1answer
21 views

Prove that any monotonic and bijective function is an homeomorphism with the usual topology

Let $f:(\mathbb{R},d_u\longrightarrow(\mathbb{R},d_u)$ an arbitrary function. Where $d_u$ is the usual distance. I have an exercise in which, with the sole asumptions of it being monotonic ...
0
votes
1answer
34 views

Proving that an arbitrary function is continuous (via topological methods)

Consider the following applications: $$f:(X,d)\longrightarrow(\mathbb{R^2},d_u)$$ $$\pi_i:(\mathbb{R^2},d_u)\longrightarrow(\mathbb{R},d_u)$$ Where $f$ is an arbitrary function, $\pi_i$ is the ...
0
votes
0answers
82 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
1
vote
3answers
92 views

Continuous map from the ring on the unit circle

Is there a surjective continuous map from the ring $r<x^2+y^2<1\,(0<r<1)$ on the unit circle $x^2+y^2<1$ ? It seems NO, but how can it be done ? Edit: what if we add the ...
2
votes
1answer
26 views

Relation between continuity and weak star continuity

Let us have a mapping $T:X^*\to X^*$. We can endow domain and codomain with norm 'strong' topology (let X be Banach space), or weak star topology. That gives us total 4 combinations: weak*-weak* ...
2
votes
1answer
59 views

Topological definition of continuity and its application to epsilon-delta definition?

So I am beginning Munkres' textbook on topology. The topological definition of continuity reads: $f:X\rightarrow Y$ is continuous if for each open subset $V\subset Y$, $f^{-1}(V)$ is an open subset ...
3
votes
2answers
33 views

How to “convert” from net to sequence in a first countable space

In a first countable space, what's a good way of going from nets to sequences? Let me explain more clearly what I mean. Suppose $f:X\to Y$ is a topological map and $X$ is first countable. Then I ...
0
votes
1answer
40 views

If $f$ and $g$ are continuous, prove or disprove that the set $\{x \in \mathbb{R} : f(x)\le g(x)\}$ is closed

Let $f,g : (\mathbb{R};J_s)\to (\mathbb{R};J_s)$, (where $J_s$ is the usual (standard) topology on $\mathbb{R}$) be continuous. Prove or disprove: (a) the set $\{x\in \mathbb{R} : f(x)\le g(x)\}$ ...
1
vote
1answer
42 views

Is a function continuous iff its restriction to each element of an open cover is continuous

Let $(X;T_1)$ and $(Y;T_2)$ be topological spaces and let $A$ and $B$ be nonempty subsets of $X$ with $A\cup B= X$ Suppose $f:X\rightarrow Y$ is a function. Then prove or disprove: (a) if $f_A$ and ...
0
votes
1answer
35 views

Let $f: X \rightarrow [0,1]$ where $f^{-1}([0,a))$ and $f^{-1}((b,1])$ are open sets in $X$ prove $f$ is continuous

Let $f: X \rightarrow [0,1]$ where $f^{-1}([0,a))$ and $f^{-1}((b,1])$ are open sets in $X$ for each $0<a,b<1$ prove $f$ is continuous The problem with question it's not clear which topology ...
0
votes
0answers
33 views

Continuity of a function in the product topoogy

Hi everyone I would like to understand if my reasoning is correct. Let $X$ be the space of sequences with values in the interval $[0,1]$, i.e. if $\mathbb{N}$ is the set of natural numbers, $x\in X$ ...
3
votes
0answers
37 views

When the inverse image of an *open* set is *closed*

Let $X$ and $Y$ be topological spaces. Assume that $f\colon X\to Y$ satisfies that the inverse image of any open set in $Y$ is closed in $X$ (as opposed to the definition of continuity). Can anything ...
1
vote
1answer
21 views

How to check if this function is semicontinuous

Could you tell me how to check that this functions are semicontinuous? $(X, \tau)$ - topological space, $ \ X \neq \emptyset$, $ \ f: X \rightarrow \bar{\mathbb{R}}$, $ \ \bar{\mathbb{R}} = [- ...
1
vote
1answer
32 views

Is $[0,1]^{[0,1]}$ Hausdorff and first-countable?

I'm trying to determine if $[0,1]^{[0,1]}$ is Hausdorff or first-countable. What I know until now, is that $[0,1]^{[0,1]}$ has the product topology, then if $x\in [0,1]$ and $U$ open in $[0,1]$ the ...
3
votes
1answer
41 views

What does it means that sequences characterize closed sets and functions?

A text book I'm reading says at one point the following: "In metric spaces are sequences the ones which chacterize closed sets and continuous functions". What is exactly the meaning of that ...
0
votes
0answers
19 views

Neighborhoods for continuous functions between CG spaces

I have a couple of problems regarding the existence of certain neighborhoods, so as to prove continuity of suitable functions. Suppose then that $Y,X$ and $Z$ are compactly-generated Hausdorff spaces ...
1
vote
1answer
59 views

Proofs about continuity and convergence in topological spaces

I'm working on the following exercise: Let $f:(X,T)\to(Y,S)$ and $x\in X$. Prove that if $f$ is continuous at $x$ then if a sequence $\{x_n\}$ converge to $x$ we have $f(\{x_n\})\to f(x)$, show ...
2
votes
1answer
50 views

Finite cyclic subgroups of the homeomorphisms of the real numbers

This inquiry was inspired by this post: The automorphism group of the real line with standard topology. While trying to come up with some interesting examples, I attempted to figure out the finite ...
0
votes
1answer
25 views

Proving the bijectivity and continuity of a function.

Let $X=[0,1]\cup(2,3]$ and $Y=[0,2]$ with the usual topology. Define $f:X\to Y$ by $$f(x) = \left\lbrace \begin{array}{l} x &\text{ if } x \in[0,1] \\ x-1 &\text{ if } ...
0
votes
0answers
14 views

A question in the Maximum and Minimum Value Theorem

This is an excerpt from Bartle's The Elements of Real Analysis. I'm having trouble trying to understand the second sentence of the proof. What does $f(x_n)\ge M-\frac{1}{n}$ show exactly?
0
votes
2answers
17 views

Need help explaining something involving Global Continuity

I'm trying to understand why the Corollary of the Global Continuity Theorem does not contradict the statement made after it.
1
vote
0answers
33 views

A bijective function $f$ between two compact Hausdorff spaces is continuous if $f$ preserves compact sets [duplicate]

I am trying to prove that if $f: X \longrightarrow Y$ is a bijection between two compact Hausdorff spaces such that $f[W]$ is compact in $Y$ for all compact $W$ in $X$, then $f$ is continuous. Here ...
0
votes
4answers
69 views

A continuous surjective function from $(0,1]$ onto $[0,1]$

I'm trying to construct a continuous surjection from $(0,1]$ onto $[0,1]$, but I'm not getting anywhere. I don't immediately see a contradiction which falsifies the existence of such a function, so my ...
1
vote
2answers
38 views

Need explaining on aspect of a proof

So the textbook I'm reading just stated three definitions of continuity: a) f is continuous at a (using neighborhoods) b) the epsilon-delta definition c) If $x_n$ is a any sequence of elements of ...
0
votes
1answer
29 views

Function continuous Uryson's lemma?

when we proved Uryson's lemma we checked that the function $f:X \rightarrow [0,1]$, where $X$ is a $T_4$ space, i continuous by checking whether $f^{-1}([0,a))$ and $f^{-1}((b,1])$ are open. $f$ is ...
1
vote
0answers
22 views

Compactification via embeddings and extending continuous functions

My question comes from reading Munkres' Topology, the section on Stone-Čech compactification. To find the compactification $\mathrm{Y}$ of $\mathrm{X}$, we find an embedding h, $\mathrm{h}: X ...
0
votes
1answer
32 views

Homeomorphism between spaces equipped with cofinite topologies

I was given this question on my midterm. Currently I am studying for finals and am still unsure how to properly solve this question. Let X and Y be two sets and f be a map from X to Y be a bijection. ...
6
votes
0answers
52 views

Axiomatizing topology through continuous maps

Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...
4
votes
2answers
49 views

For $f$ a continuous topological mapping, when are the values on the boundary of a set determined?

Suppose $f:X\to Y$ is a continuous map between topological spaces, and suppose we know the value of $f$ on a subset $S\subset X$. Continuity tells us that $f(\bar{S})\subset \overline{f(S)}$ for any ...
0
votes
0answers
33 views

upper hemicontinuity

Let $g: \mathbb R^2_+ \to \mathbb R_+$ and $h: \mathbb R^2_+ \to \mathbb R_+$ continous functions. For every $ t \in \mathbb R_+$, 1) $g(t, \cdot)$ has a unique maximum at $V(t)$ where $V: \mathbb ...
1
vote
2answers
51 views

Why is this subset not open?

I have a function, $f:[0, 1) \rightarrow \mathbb{S}^1$ given by $f(x) = (\cos2\pi x, \sin2\pi x)$. I have to show that $f$ is bijective and continuous and that $f^{-1}$ is not continuous. I have ...
0
votes
0answers
40 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
0
votes
1answer
55 views

Continuity of the sum of continuous functions

Let $X$ be a topological space and $f:X\to \mathbb{R}$ and $g:X\to \mathbb{R}$ be continuous functions. How do I show that $h:X\to \mathbb{R}$ where $h:=f+g$ is continuous, would prefer to use the ...