# Tagged Questions

53 views

### A question on the purpose of the condition on hausdorff to prove homeomorphism

This is a theorem proved in Munkres. Let $f:X\to Y$ be a bijective continous function. If X is compact and Y is hausdorff, then f is a homeomorphism. I knew Y being hausdorff which will be good to ...
87 views

### Continuous function from R to a compact set

I know that a continuous function maps compact sets into compact sets. My question now is, are there continuous functions $f:{\mathbb R}\rightarrow I$, with $I=[a,b]$ ($a\neq b$)?
34 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: f(y) = \begin{cases} 0, &l(y)<\rho \\ ...
48 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 $\epsilon>0$ ...
57 views

### Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
34 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 ...
147 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 ...
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 ...
24 views

42 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 ...
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 $d(f(x),f(y))= d(x,y)\, \forall x,y\in K$? ...
398 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 ...
94 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. ...
80 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 ...
246 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 ...
56 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 ...
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 ...
170 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 ...
35 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 ...
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 ...
57 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 ...
106 views

### The Arzelà–Ascoli theorem fails on a half-open interval

Can we find an example: (1) $\lbrace f_n \rbrace_n$ is a family of real-valued functions defined on $[0,1)$ such that this family is uniformly bounded and equicontinuous, $f_n(0)=0$; ~~~ Uniformly ...
55 views

256 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 ...
395 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 ...
151 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 ...
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. ...
256 views

### Prove or give a counterexample to the following converse of theorem: A continuous function on a compact set K(subset R) is uniformly continuous.

I think the converse of this theorem is: if every continuous function over $K$ is uniformaly continuous, then $K$ is compact. To find a counterexample of it, I want to show there exist a continuous ...
Let $X$ be a metric space. Show that all continuous functions from $X$ to $\Bbb R$ are bounded iff $X$ is compact.
### 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 ...