0
votes
2answers
53 views

How to show $g:A\to f(A)$ is open?

Let $ (X,\tau) $ and $(Y,\tau^*)$ topological spaces, $A\in \tau$ and $f:X\to Y$ is open function. Show $g:A \to f(A)$ which is generated by restricting of function $f$, is open. Let $g:X\to ...
0
votes
3answers
64 views

Example of a function?

$f$ is a discontinuous and bounded function defined on a closed set $C$. Also there exists a non-discrete closed subset in the image of $f$ such that it's inverse is open. Can you give an example ...
2
votes
2answers
27 views

How to show $f(\Bbb R,\tau_e)\to(X,\tau)$ is continuous?

Let $\tau =\{\emptyset,X,\{a\},\{c\},\{a,c\}\}$ on $X=\{a,b,c\}$ and $\tau_e$ is usual topology on $\Bbb R$ $$f(x)=\begin{cases}a,\;\;x<0\\b,\;\;0\leq ...
1
vote
3answers
73 views

Alternative to epsilon-delta definition

Back in Multivariable Calculus, I remember my professor, when explaining why he decided to skip a rigorous definition of limit (the reason was that those of us that would continue in math would go ...
2
votes
0answers
28 views

example of a regular Y such that F(X,Y) with open-compact topology is not regular? [closed]

this question from (elementary topology by s.willard ) page 288 give an example of a regular Y such that F(X,Y) (space of function not space of continuous function) with open-compact topology is not ...
1
vote
1answer
28 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
3
votes
1answer
51 views

Weak Convergence of a Sequence of Functions.

To begin my question I wish to first clarify the definition of weak convergence FOR a sequence of functions. We say that given sequence of functions, $\{f_{n}\}_{n=1}^{\infty}$, such that each $f_n$ ...
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 ...
2
votes
1answer
32 views

Show that $S^1 - \lbrace (1,0)\rbrace$ is homeomorphic to the open interval $(0,1)$

Be $S^1$ the unit circle in the plane, that is, $S^1= \lbrace (x,y) : x^2+y^2=1 \rbrace$ with the subspace topology. Show that $S^1 - \lbrace (1,0)\rbrace$ is homeomorphic to the open interval ...
0
votes
0answers
41 views

A lemma on function spaces

This is a lemma about function spaces. I'm not really understanding it however. Can someone try explaining it to me? Lemma: let $X$ be in |SET| $(Y, d)$ in |MET|, $f_n$, $f$ is in $Y^X$. Then $f_n\to ...
0
votes
2answers
37 views

Relationship between $f$ and $f^{-1}$ unclear

Say I have the function from $f:X \to Y$, $f(x) = 3$ when $x \ge 0, \ = 0$ when $x < 0$. $X$ and $Y$ both with the standard topology. Hence $f^{-1}(y) = [0, \infty)$ when $y = 3, = (-\infty, 0)$ ...
1
vote
0answers
27 views

Generating sets of topology on $C_b(\mathbb T)$ with supremum Norm?

Can someone tell me what sets generate the topology on the continuous bounded function on the 1-dim Torus $C_b(\mathbb T)$ with the supremum norm? Are the sets $\left\{ f \in C_b(\mathbb T) : ...
1
vote
2answers
95 views

Questions about continuous functions.

Recently when working with my thesis, I've got 2 questions. Let $S_n$ be the set $\{x=(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n\mid x_1+x_2+\cdots+x_n=1~\mbox{and}~0\leq x_i~\mbox{for}~ i=1,2,\cdots,n\}$. ...
0
votes
4answers
59 views

Connected Subsets of X x Y

Let $X$ be a connected topological space and $f : X\to Y$ a map. Show that the graph $G(f)$, defined by $G(f) = \{(x; f(x)) \in X \times Y | x \in X\}$ is a connected subset of $X \times Y$. I ...
4
votes
1answer
85 views

Different versions of Urysohn's Lemma

In short, I encountered two different versions of Urysohn Lemma, which I shall present here: Version 1: [Big Rudin, 2.12, page 39]: Suppose X is a locally compact Hausdorff space, V is open in X, $K ...
1
vote
0answers
29 views

Convergence of a Sequence of Functions

The context: verifying the group axioms for the fundamental group, specifically that every element has a unique inverse. Below is a non-example, and I am tasked with explaining $why$ it fails. Let ...
0
votes
2answers
65 views

Proof a function is continous.

Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, where $f(x_1,x_2) = x_1^2 + x_2^2$, a continuous function? My attempt: Suppose that $\forall \varepsilon > 0$ $\exists \delta >0$ such ...
0
votes
1answer
45 views

Inverse image of codomain

Is it true that if $f: X \rightarrow Y$ is continuous function then $f^{-1}[Y] = X$ ? I suppose that it is true and I try prove it: Suppose that $f^{-1}[Y] \neq X$ so exsists $x$ such that $ x \in ...
0
votes
2answers
46 views

Proof this function is constant

I have the following topological space: $\tau= \{U\subseteq R: 1\notin U\} \cup \{R\}$ and the following application: $f: (R, \tau)\to (R, \tau)$ I have already proved that if $f(1)=1$, then $f$ ...
0
votes
1answer
104 views

Is the function $f(x,y)=\begin {cases} ( x^3(i+1)-y^3(1-i))/(x^2+y^2) & x^2+y^2>0\\0 & x=y=0 \end{cases}$ continuous?

Is the function $f(x,y)=\begin {cases} \frac{ x^3(i+1)-y^3(1-i)}{x^2+y^2} & x^2+y^2>0\\ 0 & x=y=0 \end{cases}$ continuous? I would like to prove that thos function isn't continuous with ...
4
votes
0answers
108 views

Lower semicontinuous and discontinuous everywhere real bounded function?

Does there exist an $f:\mathbb{R}\rightarrow\mathbb{R}$ that is bounded such that for any $a$ then $f^{-1}(a,+\infty)$ is open but $f$ is discontinuous everywhere? Such a function seems too likely to ...
2
votes
0answers
42 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
3
votes
2answers
89 views

A function continuous in both arguments

Is there a two-arguments function which is not continuous but continuous in each argument? It seems I have studied something like this, but don't remember.
4
votes
2answers
91 views

Doubt in my proof (basic topology)

The exercise was to prove there exists no surjection from a connected space to $S^0$. I'm not particularly satisfied with the end of my proof though, and worse, I'm sure it's a rather stupid question! ...
1
vote
1answer
60 views

Is every continuous closed surjection also open?

$f:X\rightarrow Y$ is a continuous closed surjection, $X$ and $Y$ are topological spaces. Is $f$ also open?
0
votes
3answers
90 views

If $W\subset X$ is an open set, is $f^{-1}(Y-f(X-W))\subset W$ for any closed map $f:X\to Y$?

Let $f:X\to Y$ be a closed map. My book "Topology and Geometry" by Bredon says $f^{-1}(Y-f(X-W))\subset W$, where $W\subset X$ is an open set. Shouldn't $f$ be declared to be surjective for ...
1
vote
2answers
85 views

Defining a function via restrictions of many bijections

I'm trying to define a certain surjective map from $(0,1) \to (0,1)$. For any $(a,b) \subseteq [0,1]$, there's a bijection $(a,b) \rightarrow (a,b) \cup (\frac{1}{3},\frac{2}{3})$. Is there a way to ...
3
votes
2answers
95 views

Two equal functions on a topological space

Can anybody help please help me, I have to answer this problem in topology: "Let $f$ and $g$ be continuous functions from the topological space $T$ into $\mathbb{R}$, with the usual topology. Show ...
5
votes
3answers
119 views

Open, closed and continuous

I have some troubles to understanding something: We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example ...
2
votes
1answer
61 views

When in topology is $A = f^{-1} \circ f[A]$ or $B = f \circ f^{-1}[B]$ true, for an $f$ which is not one-to-one?

I'm having a bit of trouble with an example problem in the topology book I'm reading. It's problem #11 (pp 104) of the "Solved Problems" section of Chapter 7, of the Schaum's Outline for "General ...
1
vote
2answers
35 views

Homeomorphism $id_M:(M,\tau_d)\rightarrow(M,\tau_h)$

I am reading thorugh some topological definitions, in my book it is stated that $id_M:(M,\tau_d)\rightarrow(M,\tau_h),x\rightarrow x$ is a Homeomorphism where $(M,d)$ is a metric space, ...
5
votes
2answers
127 views

If $f$ and $g$ are continuous, prove $f\circ g$ is continuous.

Suppose that $(X,T)$, $(Y,U)$ and $(Z,V)$ are three topological spaces and that $g\colon X\to Y$ and $h\colon Y \to Z$ are continuous. Prove that $h\circ g\colon X \to Z$ is a continuous ...
0
votes
1answer
49 views

Showing a certain function is open and not closed

I'm trying to show that $f:\mathbb{R}\to S^{1}$ ($S^{1}$ being the unit sphere in $\mathbb{R}^{2})$ defined by $$f\left(t\right)=\left(\cos\left(2\pi t\right),\sin\left(2\pi t\right)\right)$$ ...
0
votes
0answers
39 views

Small question regarding open mappings

A function $f:X\to Y$ between topological spaces is said to be open if for every $U\subseteq X$ which is open $f\left(U\right)\subseteq Y$ is open. I'm trying to show that a function is open ...
3
votes
1answer
317 views

Application of Banach fixed-point theorem

I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satifies $\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$ for $t\in[0,1]$. The first thing I do is to define a function ...
4
votes
3answers
140 views

A description of all continuous functions on the quotient space.

Is there any method which allows us to describe all continuous functions (maps to $\mathbb{R}$) on the quotient space? For examle, how could I classify all continuous functions on ...
1
vote
0answers
71 views

Continuity and openness of the map $C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$

I need to prove or disprove that the composition operator is continuous and open. Consider the following map $$h:C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$$ that takes a function ...
2
votes
1answer
154 views

Prove that a given subspace of $C[-1,1]$ with $L^2$ norm is closed

Let $H= C[-1,1]$ with $L^2$ norm and consider $G=\{f \in H \mid f(1) = 0\}$. Show that $G$ is a closed subspace of $H$. I've been trying to prove this for a while but i can't establish that given ...
3
votes
1answer
444 views

Homeomorphism between $S^1$ and $[0,1]/ 0\sim1$

I'm studying quotient spaces in my topology course. I want to prove that circle $S^1=\{(x,y)\mid x^2+y^2=1 \}$ homeomorphic to $[0,1]/ 0\sim1$ (segment with the identified points). To solve this ...
2
votes
2answers
94 views

questions on a continuous, injective, surjective

Let $f: X\rightarrow Y$ be a continuous, injective, surjective. Question 1, if $f$ is open or closed, then does $f^{-1}$ continuous? Question 2, if $f$ is open or closed, then does $f^{-1}$ open or ...
0
votes
1answer
62 views

A question on $C_p(X)$

What is the sufficient condition on $X$ which makes the spce $C_p(X)$ is the first countable? Thanks!
2
votes
1answer
176 views

Existence of bump functions which are positive on a prescribed set

Let $U \subset \mathbb{R}^n$ be an open subset of Euclidean space. I feel like there should be a smooth function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with $f|_U > 0$ and ...
0
votes
1answer
219 views

Does a function sequence decreasing monotonically to 0 converge uniformly?

Suppose $\{f_n\}$ be a sequence of continuous function$f_n:S\to \mathbb{R}$ where $S\subset \mathbb{R}$ and $S$ is compact. Suppose for $\{f_n(x)\}$ monotonic decreasing to zero for any $x\in S$. Is ...
3
votes
2answers
737 views

Extreme points of unit ball in $C(X)$

Let $X$ be a compact Hausdorff space and $C(X)$ be the space of continuos functions in sup-norm. I read in Douglas' Banach algebra techniques in operator theory that the followings are equivalent: ...
1
vote
2answers
121 views

Continuity at an arbitrary point

Let $f:\Bbb N \to\Bbb R$ by writing $f(n) = \frac1{n^2}$. Is $f$ continuous at any point in its domain. So, my thought is, $f$ is a function with domain $\Bbb N$ - natural numbers, so each point ...
1
vote
0answers
45 views

Manifolds question

Let $M$ subset of $R^{n+p}$ be the zero set of a $C^\infty$ mapping $g:R^{n+p} \rightarrow R^{p}$. Assume that the Jacobi matrix of $g$ has rank $p$ everywhere on $M$. Show that $M$ is an ...
2
votes
2answers
150 views

True statements for a continuous function

Let $f\colon \mathbb R\rightarrow \mathbb R$ be a continuous function. Define $G = \{(x, f(x)) : x \in \mathbb R\} \subseteq \mathbb R^2$. Pick out the true statements: a. $G$ is closed in $\mathbb ...
5
votes
1answer
600 views

How to prove the function $y = \sin x$ is not a closed function?

I came across a question: Suppose that $f(x) = \sin x$ is a function from $\mathbb R$ to $[-1,1]$. How do I prove the function $f(x) = \sin x$ is not a closed function? By "closed function", ...
5
votes
2answers
75 views

Can such a function exist?

Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$. Does there exist a function $f: \mathbb{R} \to ...