1
vote
1answer
33 views

Are the family of given nice functions $f\subset C^0(I,[0,1])$ equicontinuous?

The family of continuous functions $f\in\mathcal{F}$ are defined on a closed subset of real numbers $I\subset\mathbb{R}$ as follows: \begin{equation} f(y) = \begin{cases} 0, &l(y)<\rho \\ ...
0
votes
1answer
44 views

Are the family of functions $C^0(I,[0,1])$ equicontinuous?

I searched but couldn't find. Are the family of continuous functions $C^0(I,[0,1])$ equicontinuous for the finite interval $I\subset\mathbb{R}$? To claim this, I guess for every $\epsilon>0$ ...
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!
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 ...
4
votes
2answers
143 views

Map preserving intervals but discontinuous

Let $f:\mathbb{R}\to\mathbb{R}$ be a map sending closed intervals to closed intervals. Prove that $f$ is continuous or find a counter example. WLOG we just have to prove continuity at $0$ and we can ...
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 ...
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 ...
3
votes
1answer
40 views

Proof of compactness of Lipschitz functions

Consider the set $\mathcal{F}$ of continuous functions on $[0;1]$ with boundary values $$ f(0)=f(1)=0 \qquad \forall f \in \mathcal{F}. $$ Define the metric $d(f,g) = \lVert f-g \rVert_\infty = ...
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 ...
3
votes
0answers
24 views

Surjective function on a compact metric space [duplicate]

Assume $f:K\rightarrow K$, is surjective and $K$ is a compact metric space and we have $d(f(x),f(y))\leq d(x,y)\, \forall x,y\in K$. How can I prove that $d(f(x),f(y))= d(x,y)\, \forall x,y\in K$? ...
3
votes
3answers
375 views

Inverse image of a compact set is compact

Let $X$ and $Y$ be topological spaces, $X$ compact, $f : X \to Y$ continuous. Then the preimage of each compact subset of $Y$ is compact. With the stipulation that $X$ and $Y$ are metric spaces, this ...
2
votes
0answers
91 views

Counterexample to Converse of Extreme Value Theorem?

The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. ...
2
votes
2answers
77 views

Compact Domain and Inverse Image

I am trying to show that given $f:M \rightarrow N$, where $M$ is compact, $f$ is continuous and onto, then given $A \subset N$: $$ f^{-1}(A) \text{ closed} \implies A\text{ closed} $$ I am dealing ...
1
vote
2answers
244 views

Compactness implies Continuity?

I am stuck on this question (probably there are many counterexamples, but I can't find any). "Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ that preserves compactness (i.e, for every $K \subseteq R$, then ...
0
votes
3answers
55 views

Compact subsets of a metric space

I am trying to to prove that f: X --> Y is continuous on X if and only if f is continuous on every compact subset of X. X and Y are metric spaces. How do I show that every point of X belongs to some ...
1
vote
2answers
42 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
5
votes
2answers
168 views

Topology problem - compactness

How to solve the following: Let $X$ be a locally compact, $Y$ Hausdorff space and $f : X\rightarrow Y$ continuous open surjection. Prove that for every compact set $K\subset Y$ exists compact set ...
0
votes
1answer
34 views

Uniformly continuous on a compact set, still uniform on a subset?

So if I have a function that is uniformly continuous on a compact set K, do all subsets of K inherit the uniform continuity? If I restrict myself to the reals, this seems to be true. But what happens ...
1
vote
1answer
64 views

convergence of compact sets

Let $X$ and $Y$ be compact sets (both subsets of the real line). Assume $Y$ has a non-empty interior. Consider the continuous function $f:X \rightarrow Y$. For any given $y$ in the interior of Y, and ...
2
votes
1answer
55 views

Convergence of compact sets continuous function

Let $X$ and $Y$ be compact set (both subset of the real number). Consider the continuous function $f:X \rightarrow Y$. For any given $y$, and for $h>0$ small enough so that $y+h \in Y$. I want to ...
2
votes
1answer
55 views

Function for which taking preimages preserves limit points

Suppose we have a surjection $f : X \to Y$ between topological spaces. What is the weakest assumption on $f$, $X$ and $Y$ you can think of that endows $f$ with the following property? If $A ...
0
votes
2answers
84 views

compact subset and proof

$X$ be a compact subset of $\mathbb{R}$ and let $f$ be a real-valued function on $X$. Prove that $f$ is continuous if and only if $\{(x,f(x)) : x ∈ X\} $ is compact subset of $\mathbb{R}^2$. how ...
3
votes
1answer
204 views

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum. I want to prove this. This is my proof: Since $X$ ...
1
vote
4answers
121 views

Does extreme value theorem hold when continuous is replaced with bounded?

The extreme value theorem says that if the domain of a 'continuous' function is compact then both the max and min of the function lies in the domain set. My question is: can the 'continuity' be ...
0
votes
1answer
62 views

Prove that closure of $f(U)$ is compact in $Y$.

Let $f: X \to Y$ be continuous, where $X$ is compact and $Y$ is Hausdorff. Let $K$ be a closed subset of $X$ and $U$ a proper subset of K. Prove that closure of $f(U)$ is compact in $Y$. I have got ...
7
votes
2answers
897 views

True Or not: Compact iff every continuous function is bounded [duplicate]

Let $X$ be a topological space. My question is: If $f:X\to \mathbb{R}$ is bounded for all such continuous $f$, then is $X$ compact. Is is really? If $X$ is the subset of $\mathbb{R}^d$, then it is ...
3
votes
1answer
380 views

Compactly supported continuous function is uniformly continuous

Let $f:\mathbb R \rightarrow \mathbb R$ be continuous and compactly supported. How can I prove that $f$ is uniformly continuous ? I was trying to prove it by contradiction but get stuck. My attempt ...
2
votes
2answers
73 views

Continuous function from non-compact space onto compact space

Give an example of metric spaces $M_1$ and $M_2$ and a continuous function $f$ from $M_1$ onto $M_2$ such that $M_2$ is compact, but $M_1$ is not compact. So there must exist a sequence ...
0
votes
1answer
47 views

Number of Discontinuities of a Monotone function of several variables

We know that any monotone function from $\mathbb{R} \to \mathbb{R}$ has only countable discontinuities. What about monotone functions on, say $[a,b]^2 \to [a,b]^2$ (i.e., component-wise monotone)? ...
4
votes
1answer
102 views

Continuity on a product space, one of which is compactly generated

Let $A$ and $B$ be Hausdorff topological spaces, with $A$ compactly generated (i.e., a subset of $A$ is open iff its intersection with each compact subset $K$ of $A$ is open in $K$; from 2nd edition ...
1
vote
2answers
312 views

Finding a bounded, non-compact set of functions $f:[0,1]\to\Bbb R $

Consider the metric space $(X, d)$ given by $$X = \{\text{all continuous functions}\,f:[0,1]\to\Bbb R\}$$ with $$d(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|.$$ Find with proof a set $A \subseteq X$ with ...
2
votes
1answer
56 views

A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
3
votes
1answer
76 views

How to show that a continuous map on a compact metric space must fix some non-empty set.

Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$ I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
2
votes
1answer
68 views

exercises in compactness

I am working on some practice problems on Compactness. (Q.1.a Chapter 1.7 in Advanced Calculus, Folland) The question is : Give an example of : a closed set $S\subset R\quad$ and a continuous ...
1
vote
1answer
63 views

Compact metric space: proof $\text{diam}(K)$

I am to assume that $K$ is a compact metric space. I must prove that there are two points $x,y$ contained in $K$ such that $d(x,y)=\text{diam}(K)$. Recall $\text{diam}(K)= \sup \{ d(x,y) \mid x,y ...
2
votes
1answer
254 views

$f:M_1\to M_2$ is continuous iff its graph is compact.

I have a propostion in Introduction to Real Analysis (3rd Ed.) which says: If $M_1$ is compact, a function $f:M_1\to M_2$ is continuous iff its graph is compact. Here $M_1$ and $M_2$ are ...
1
vote
2answers
385 views

Bounded functions on subsets of Euclidean space

It is known that given any closed and bounded $X \subseteq \mathbb{R}^n$ and a bounded continuous function $f : X \to \mathbb{R}$, $f(X)$ has a minimum value and maximum value. This can be proved by ...
0
votes
1answer
150 views

Yes or No, Real Analysis, continuity, compactness

Am I correct over statements below? The limsup and liminf of the sequence $n^2$ (meaning $1,4,9,16,\dots$) are equal. T Every bounded sequence has at most one ...
1
vote
1answer
284 views

Uniform convergence of continuous functions with Lipschitz limit

Let $K \subset \mathbb R^d$ be a compact. Let $\phi_{\varepsilon} \colon K \rightarrow \mathbb R$ be continuous and converge uniformly to $\phi$. Suppose further that $\phi$ is Lipschitz continuous. ...
2
votes
2answers
249 views
1
vote
1answer
128 views

About necessary and sufficient condition in compact metric space

Let $X$ be a metric space. Show that all continuous functions from $X$ to $\Bbb R$ are bounded iff $X$ is compact.
6
votes
1answer
115 views

Continuous functions on $\beta\mathbb{N}$.

Let $f\colon \beta \mathbb{N}\to \mathbb{C}$ be a continuous function such that $f(x)=0$ for some $x\in \beta\mathbb{N}\setminus \mathbb{N}$. Can we conclude that there exists both closed and open ...