1
vote
0answers
27 views

Sequence of increasing compact sets

Suppose $X$ is a locally compact metric space which is $\sigma$-compact. Let $K$ be a compact subset of $X$. We can find a sequence of compact sets $K_{n}$ such that $K_{n} \subset \textrm{int}(K_{n + ...
2
votes
1answer
35 views

The set $S=\{(x,y) \in \mathbb{R}^{n} \times \mathbb{R}^n = \mathbb{R}^{2n} ; x \neq y\}$ is connected if $n \geq 2$.

When n = 1 it is easy to see that is not connected, it just take the split open $ S=U_1 \cup U_2$ such that $U_1 = \{(x,y) \in \mathbb{R}^2 ; x > y\}$ is $U_2 = \{(x,y) \in \mathbb{R}^2 ; x < ...
0
votes
0answers
25 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 ...
7
votes
2answers
193 views

The phrases “has … in ” vs. “contains … of” in Baby Rudin

Consider the following two statements. (Assume $E \subseteq K$.) $E$ has a limit point in $K$. vs. $E$ contains a limit point of $K$. What do they each mean and how are they different?
2
votes
2answers
49 views

Derivatives on Functors

I'm not even sure if this question makes pedantic sense but is there any way to rigorously define the notion of taking the derivative of a functor?
3
votes
1answer
33 views

translation of open set problem

Suppose U is an open set in an Euclidean space. Then any point in U is contained in all but finitely many open sets that is translated by vectors converging to zero. It is easily proved in one ...
2
votes
0answers
31 views

Prove that $\mathbb{N}$ with cofinite topology is not path-connected space.

$\mathbb{N}$ is the set of natural numbers. Let $U_{\alpha \in A} \subset \mathbb{N}$ be the subset such that its complement $\mathbb{N}$ \ $U_\alpha$ is a finite subset. Then $T= \{\emptyset, ...
0
votes
1answer
20 views

Prove that $\mathbb{N}$ is not metrizable where $U$ is open if it is $U=\mathbb{N}$, $U = \emptyset$, or $\mathbb{N}$ \ $U$ is a finite subset.

$\mathbb{N}$ is the set of natural numbers. Any set $U$ is open if it is $U=\mathbb{N}$, $U = \emptyset$, or $\mathbb{N}$ \ $U$ is a finite subset. This defines a topology on $\mathbb{N}$. Prove ...
0
votes
2answers
23 views

Compact subset of a closed subspace: compact in the whole space?

Imagine that you have a topological space $(X,\tau_{X})$ and a closed subset $Y$. Say that within $Y$ we have a subset $K$ that is compact in the subspace topology $\tau_Y$. Is $K$ compact in ...
0
votes
1answer
39 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
0
votes
0answers
15 views

understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...
0
votes
1answer
35 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
0
votes
1answer
26 views

Can we take the infimum over a variable set?

Suppose that we have a family of functions $\lbrace f_{\alpha}(x)\rbrace_{\alpha}$ define on an open set of $\mathbb{R}^{m}$, and $\alpha$ runs over a set $\Gamma$. Assume that the family is ...
0
votes
0answers
92 views

understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
2
votes
1answer
67 views

compactness in topology of pointwise convergence

I started reading about the topology of pointwise convergence. So far I do not feel quite comfortable with this theory. Maybe one can help me out in a more concrete example case. Let's consider ...
2
votes
4answers
63 views

A question about metrizability

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
0
votes
3answers
73 views

Closed subset of compact set is compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ...
0
votes
1answer
20 views

Bump function's support

How does this function (for instance) have compact support, if the support is the open interval (-1,1) ? http://en.wikipedia.org/wiki/Bump_function
-1
votes
0answers
22 views

How to show that $\{ \sin n : n=1,2,3,\dots \}$ is dense in $[-1,1]$? [duplicate]

Show that S is dense in [-1,1]: $$ S = \{ \sin n : n=1,2,3,\dots \} $$
0
votes
1answer
26 views

Boolean Closure and Borel sets

Denote the boolean closure of a family of sets $\mathcal S$ by $\mathcal B(\mathcal F)$, then in a metric space it is well known that $\mathcal B(\mathcal F) = \mathcal B(\mathcal G) = \mathcal ...
1
vote
0answers
29 views

Order-preserving embeddings

(Follow-up to Existence of a utility function on the reals.) Say we have a totally ordered set $X$ which has a countable, dense subset $C$. I believe we can find an $f:C\to\mathbb R$ which is ...
0
votes
1answer
46 views

Every Bounded set contained in a Compact set

In a general metric space, is every bounded set contained in a compact set?
0
votes
1answer
34 views

Is every separately continuous function on $R^2$ continuous?

My friend asked me this question a few days ago. I felt it's not right but couldn't find a single counterexample. Any comment is appreciated.
5
votes
1answer
55 views

Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
2
votes
2answers
51 views

Is this a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$. Is this a metric on $\mathbb{R}$? It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality ...
3
votes
1answer
111 views

Topologies of test functions and distributions

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my ...
0
votes
0answers
28 views

Sequence Criterion of Continuity and Divergent sequences

A function $f : (X, d_X) \to (Y, d_Y)$ between metric spaces is continuous iff $$ \lim_{n \to \infty} f(x_n) = f(\lim_{n\to \infty} x_n) $$ for each convergent sequence $(x_n)$ in $(X, d_X)$. So if I ...
0
votes
0answers
25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
0
votes
1answer
69 views

compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
0
votes
1answer
33 views

Interior Points Confusion

I am confused about interior points. Basically, I think that if the set we are not working in doesn't contain irrational numbers, then the interior of the set is $\emptyset$. $\left [ 0, \; 5\right ...
1
vote
4answers
73 views

Show the set is open

I've been seriously trying to solve this for really lots of time. It's pretty basic, which furthers the frustration... Let there be a set $A=\{x=(x_1,x_2)\in \mathbb R^2 :|x_1|+|x_2|<1\}$. ...
0
votes
3answers
63 views

Counter Example about Continuous Functions

(James Munkres page 104 Theorem 18.1) Let $X$ and $Y$ be topological spaces; let $f: X \rightarrow Y$. If $f$ is continuous, then for every subset $A$ of $X$, one has $f(\overline{A})\subset ...
4
votes
1answer
50 views

Whether a set is closed or not

Denote by $C_{[0,1]}$ the ternary Cantor set on $[0,1]$. Now consider $[0,1] \setminus C_{[0,1]}$. It contains open intervals. Now define Cantor sets on all these open intervals by simply translating ...
4
votes
1answer
61 views

How do we know ternary expansions with only $0$'s and $2$'s are unique?

Let $c \in [0,1]$ and consider one of its ternary expansions $\sum_{n \ge 1} c_n / 3^n$ s.t. each $c_n = 0$, $1$, or $2$. This ternary expansion needn't be unique. For example: $$ 0.0222222\ldots ...
0
votes
1answer
33 views

Extending $C \cong C \times C$ to the Space-Filling Curve $[0,1] \cong [0,1]^2$

Setting: Let $C$ denote the Cantor Set. I have already shown that $C \cong \prod_{i=1}^\infty \{0,2\}_i$, and make use of this fact below. Goal: Show that $C \cong (C \times C)$, and then use this ...
1
vote
1answer
25 views

A question on accumulation points

Suppose $S$ is a closed, countable, subset of $\mathbb{C}$. For any $n$, define $S_n:=S_{n-1}\backslash B_{n-1}$, where $B_{n-1}$ is the set of isolated points of $S_{n-1}$ (as usual, $S_0:=S$). Is it ...
2
votes
1answer
39 views

Showing a Space-Filling Curve is Continuous

Setting: Let $F':[0,1] \rightarrow [0,1]$ be the Cantor Function. Goal: Show that there exists a well-defined, surjective, continuous function from $[0,1]$ to $[0,1]^2$ (i.e., a space-filling curve). ...
5
votes
2answers
106 views

Compactness and closedness

If every closed and proper subset of a topological space is compact, then is the whole space necessarily compact? The "converse" of this question is well-known, of course, but I'm having difficulty ...
3
votes
1answer
109 views

Showing $f: \prod_{i \ge 1} \{0,2\}_i \rightarrow C$ is an Open Mapping into the Cantor Set

Setting: Let $C$ denote the Cantor Set and $X = \prod_{i \ge 1} \{0,2\}_i$. Let $X$ be given the product topology. Consider $f : X \rightarrow C$ s.t. if $p = \left\langle x_1, x_2 , \ldots ...
0
votes
3answers
69 views

The Product Topology

Let $X$ and $Y$ be infinite sets, let $t$ be the discrete topology on $X$ and let $u$ be the trivial topology on $Y$. Describe the product topology on $X \times Y$. This is a book problem and I was ...
1
vote
1answer
40 views

Prove that two separated sets must be relatively open in their union

I'm working in the plane here. My idea was that if A is open, then let O = A and then O intersect (A union B) = A, since A and B are separated, so A is relatively open in A union B. If A is not open, ...
1
vote
1answer
48 views

How is an open set defined without referring to any distance function in a topology?

I am currently studying general topology. The definition given by Royden looks very confusing to me. It says that the elements of a topology is called open sets without actually defining what exactly ...
2
votes
3answers
89 views

Problem in Chapter 2 (Walter Rudin).

"Principles of Mathematical Analysis" by Walter Rudin has the following question: Show that the following statement is false in $\Bbb{R}$: If $\{K_\alpha\}$ is a collection of closed subsets ...
4
votes
1answer
93 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
10
votes
6answers
608 views

Why we define the concept of continuity

The concept of continuity is a very important idea in topology. Though I am using it all the time, but indeed I don't know what is the original purpose for us to define this concept. And I also don't ...
2
votes
1answer
35 views

Continuous modification of functions with a given property

Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ with the following property: For all reals $x$, $\displaystyle\lim_{y \to x} f(y)$ exists. (In particular, note that its possible that ...
1
vote
2answers
78 views

Questions about continuous functions.

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

How to choose $f\in C^{2}(\mathbb{R})$ with compact support and takes value 1 on connected compact set?

Let $0< \delta < \pi$. My questions: (1) How to construct(choose/method) $f\in C^{2}(\mathbb R)$(= First two derivatives ($f' \ \text{and} \ f''$) of $f$ on $\mathbb R$ exists and are ...
0
votes
0answers
12 views

Countable product of metric spaces

Let $X=\prod X_i$ of countably many metric spaces $(X_i,d_i)$. Prove that the function which associates to $x=(x_i)$,$y=(y_i) \in \prod X_i$ the number $d(x,y)\in [0,\infty]$ defined by ...
0
votes
1answer
48 views

relation between topologies

I Came across this question recently. Could someone help me out with this. Let T1 be the smallest topology on $\mathbb{R}^2$ containing the sets $(a,b)\times (c, d)$ for all $a,b, c, d$ in ...