1
vote
1answer
37 views

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\cap_k B_k$ is either a point or a closed ball.

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\bigcap_k B_k$ is either a point or a closed ball. Please help me check the proof, thanks! Define $x_k$ to be ...
2
votes
2answers
25 views

Recurrent point: two definitions

Let $X$ be a topological space and $T:X\longrightarrow X$ a function. Now lets look at the two following definitions: $x\in X$ is a recurrent point if for every neighborhood $U$ (of $x$) the ...
0
votes
0answers
16 views

Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
2
votes
0answers
22 views

Prove that $d_\infty(f, g) = \operatorname{sup}\{|f(x)-g(x)|:x \in [a,b]\}$ defines a metric

Can someone please verify my proof? Let $C[a,b]$ denote the set of all continuous functions from $[a,b]$ to $\mathbb{R}$. Let $d_\infty:C[a,b] \times C[a,b] \longrightarrow [0, \infty)$ be given ...
-1
votes
1answer
43 views

Existence of certain uncountable closed sets in the order topology

This is a proof-verification request. Let $\Omega$ be the set of countable ordinals, $\omega_1$ the first uncountable ordinal, and $\Omega^*=\Omega\cup\{\omega_1\}$. Remarkable properties of these ...
0
votes
2answers
80 views

Is this NOT considered a proof?

Define the boundary of $A$ as $$Bd(A) = cl(A) \cap cl(X - A).$$ Show that $cl(A) = int(A) \cup Bd(A)$. The solution tries to show that $cl(A) \subset cl(A).$I thought I would do it directly, $int ...
1
vote
1answer
64 views

How many connected components? (CSIR June'13)

Let $X= \{ (x,y)\in \mathbb{R}^2: x^2+y^2<5\}$ and K=$\{(x,y)\in \mathbb{R}^2: 1\le x^2+y^2\le2 \quad\text{or}\quad 3\le x^2+y^2\le 4\}$ Then, 1.$X\setminus K$ has three connected ...
2
votes
2answers
36 views

My version of order topology is Hausdorff

Can someone say something about my version of "order topology implies Hausdorff" (WLOG) Let $a <b$, and let $U_1,U_2$ be a neighborhood of $a,b$ respectively. Denote $U_1 = (a - \epsilon, a + ...
3
votes
1answer
53 views

Is this right? Topology with closures

I want to show that (possibly) $$cl(A-B) = cl(A) - cl(B).$$ I know that $$cl(A-B) \subset cl(A) - cl(B).$$ already, but for the other inclusion I tried this. Let $x \in cl(A) - cl(B)$, so that for ...
0
votes
3answers
34 views

If $Y$ is compact and $f : X \rightarrow Y$ is a map whose graph $G = \{ (x,f(x) : x \in X\}$ is closed in $X \times Y$ , then $f$ is continuous.

If $Y$ is compact and $f : X \rightarrow Y$ is a map whose graph $G = \{ (x,f(x) : x \in X\}$ is closed in $X \times Y$ , then $f$ is continuous. Let $C \subseteq Y$ be a closed. Let $x \in X - ...
0
votes
2answers
27 views

Is this proof rigorous enough? Subspace of discrete space

Problem: Every subspace of a discrete space is discrete. Proof1 :Let $X$ be a discrete space with the discrete topology $\tau = 2^X$ and $Y$ be subspace with its topology $$\tau_Y = \{ Y \cap U : ...
0
votes
1answer
31 views

Prove: $\forall X \in \mathscr{I}(r)(\exists Y \subseteq X(r \in Y \wedge \forall z \in Y(Y \in \mathscr{I}(z))))$

In the book I am reading the following Prop. 1: let be $(A,B)$ a topological space, $r \in A$, and $\mathscr{I}(r):=\text{family of neighbourhoods of }r$, then $$\forall X \in \mathscr{I}(r)(\exists ...
1
vote
1answer
44 views

Determining if certian properties of a topological space pass to its image under a quotient map.

A property $P$ of topological spaces is said to "pass to quotients" if whenever $p : X \rightarrow Y$ is a quotient map and $X$ has property $P$ then $Y$ has property $P$. For the following ...
0
votes
1answer
28 views

Prove that if $p: X \rightarrow Y$ is a $2$-fold covering projection, then $p$ is a quotient map.

Prove that if $p: X \rightarrow Y$ is a $2$-fold covering projection, then $p$ is a quotient map. By the definition of a covering projection, $p$ is both continuous and surjective. We have ...
3
votes
1answer
32 views

Show that the following function is surjective and continuous but is not a quotient map.

Let $X = \Bbb{R} \times \{3, 4, ...\} \subset \Bbb{R}^2$. Let $L_{\theta} \subset \Bbb{R}^2$ be the line through the origin with slope $\tan \theta$ (i.e., the directed angle from the positive ...
4
votes
1answer
59 views

Weak topology is not metrizable: What's wrong with this proof

Let $(X,\|\cdot\|)$ be a infinite dimensional normed vector space, and Suppose that the weak topology in $X$ is metrizable by a metric $d$. How the opens of $\tau_d $ should be the same as the ...
3
votes
1answer
37 views

Deduce that the product of uncountably many copies of the real line \Bbb{R} is not metrizable.

Deduce that the product of uncountably many copies of the real line \Bbb{R} is not metrizable. Let $J$ be an uncountable set. Suppose that for $x = (x_j)_{j \in J} \in \prod_{j \in J} \Bbb{R}$ ...
1
vote
2answers
64 views

Let $f :[0,1] \rightarrow [0,1]$ be continuous with $f(0) = 0$ and $f(1) = 1$. Prove that f is onto.

Let $f :[0,1] \rightarrow[0,1]$ be continuous with $f(0) = 0$ and $f(1) = 1$. Prove that f is onto. Suppose, for contradiction, that $y \in [0,1]$ is not in the image of $f$. Since $f$ is ...
0
votes
0answers
30 views

The discontinuous or the characteristic function is the boundary

Let $A \subseteq \mathbb{R}^n$. Put $D = \{ x \in R^n : \chi_A(x) \; \; \text{is discontinuous } \} $. Then do we have that $$ \partial A = D $$ ??? My attempt: If $x \in \partial A$, then we can ...
1
vote
1answer
37 views

Prove: A set containing limit points of a sequence is a closed set

The question: Prove that a set, $S'$, containing the limit points of the sequence $S \subset \mathbb{R}$, is closed. What I have so far: I want to prove this by ...
0
votes
1answer
63 views

Prove: The union of open subsets of $\mathbb{R}$ is open

The problem is to prove that the union of open subsets of $\mathbb{R}$ is open. However, the only definitions that I have to work with are: A set is closed if it ...
0
votes
1answer
83 views

Product of weak Hausdorff space is weak Hausdorff

I have read on May's Algebraic Topology such that the category of weak Hausdorff space $\mathcal{wTop}$ has same limit as $\mathcal{Top}$, which means Product of weak Hausdorff space is weak ...
1
vote
2answers
64 views

Proving a metric induces the product topology

Let $(M,d)$ and $(N,d')$ be metric spaces. Prove that the product topology is induced by the metric $d_1((x,y),(x',y')=d(x,x')+d(y,y')$ and ...
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
0answers
51 views

Let X be a metric space in which every infinite subset has a limit point. Prove that X is compact.

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact. The following is my proof I'd like to know if it is correct. Proof: I will use the fact that ...
2
votes
1answer
50 views

Path components of Wedge Sum

I couldn't find this anywhere else, so I decided to post it here. I suspect that the wedge sum $⋁X_α$ of pointed spaces $X_α$ has as path components all components of the topological sum $\oplus X_α$ ...
2
votes
0answers
86 views

$\ f \colon X \to X $ ,continuous function where X is compact,Hausdorff space.Show $\exists A$ st $f(A) =A$.

Suppose $\ f \colon X \to X $ is a continuous function from a compact,Hausdorff space to itself. Prove that there exists a subspace $A$ such that $f(A) =A$. I came up with an answer based on nets ...
1
vote
0answers
50 views

topological equivalence on interior of $D^2$ that is not continously extendable to $D^2$

As said in the title, I'm trying to find a topological equivalence on the interior of $D^2$ that is not continously extendable to $D^2$. I have an idea about this, so here it goes: Let ...
1
vote
0answers
48 views

Why is proof of the [topological] closed graph theorem incorrect?

Specifically, the closed graph theorem I am referring to is: Let $f : X \rightarrow Y$ exist and $Y$ be compact and Hausdorff. Then $f$ is continuous if and only if the graph of $f$ denoted by $G_f = ...
0
votes
0answers
28 views

A question about open “balls”

I've been recently learning Topology and I'm struggling to visualize open balls. For instance, on $\mathbb{R}^2$ and $\mathbb{R}^3$ given a metric like say $d_\infty(x,y)=\sup\{|x_1-y_1|,|x_1-y_2|\}$ ...
2
votes
4answers
75 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 ...
3
votes
3answers
81 views

Prove $Y$ is connected

Let $A$ be a connected subspace of $X$ and suppose $A\subseteq Y\subseteq\overline{A}$. Prove that $Y$ is connected. My attempt: Suppose that $Y$ is not connected. Then $Y=U_1\cup U_2$ where $U_1$ ...
0
votes
1answer
63 views

A closed subspace of a locally compact Hausdorff space is also a locally compact Hausdorff space.

Let $X$ be a locally compact Hausdorff space, and $A$ a closed subspace. Show that $A$ is a locally compact Hausdorff space. Here is what I have for a proof. Will I need to clarify anything else? ...
0
votes
1answer
35 views

Closed subspace of a compact topological space is compact

Let $X$ be a compact topological space, and $A$ a closed subspace. Show that $A$ is compact. How does this look? Proof: In order to show that $A$ is compact. We need to show that for any open ...
0
votes
1answer
32 views

is a closed and bounded set in $\mathbb{C}^n$ under the metric induced by the standard inner product, compact?

I really do ask a question in this post, not saying that i know something. I want to clarify what i think, which may be trivial to you readers, but which I'm not really sure. I know the Heine-Borel ...
0
votes
0answers
28 views

Show that the cantor set is self similar

Alright so here is what I have: $$\sum_{n=0}^\infty \frac{1}{3}\frac{2}{3}^{n-1} = \frac{1}{3}\sum_{k=0}^\infty\frac{2}{3}^{k}$$ $$\sum_{n=0}^\infty r^{k} = \frac{1}{1-r}, r\lt 1$$ ...
2
votes
0answers
45 views

In a complete metric space $(X,\rho)$, show that if $E$ and $X\setminus E$ are dense, then at most one of them is a countable union of closed sets.

The problem statement is in the title. I approached this proof using contradiction. My attempt was: Suppose that both $E$ and $X\setminus E$ are dense and that both are a countable union of closed ...
3
votes
1answer
120 views

Help proving Cantor Intersection Theorem using Bolzano-Weierstrass Theorem

Came across the following exercise in Bartle's Elements of Real Analysis and am quite unsure about my solution. Would greatly appreciate it if someone could take a look at it. The Bolzano - ...
0
votes
1answer
43 views

Is my understanding of limit point compactness correct with respect to $[0,1]^{\omega}$ with the uniform topology?

The following is an exercise problem about limit point compactness from the book "Topology" by Munkres (2nd edition). Exercise 1 in Section 28: Give $[0,1]^{\omega}$ the uniform topology. Find an ...
2
votes
2answers
189 views

$A \times B$ is an open set in $\Bbb R^2 \implies A$ and $B$ are both open in $\Bbb R$; $A,B \neq \emptyset$

I am studying Analysis on my own. Reading The Elements of Real Analysis by Bartle. Came across the above problem and I came up with the following solution but am very unsure about it. Would be very ...
2
votes
1answer
97 views

Show that any continuous map $f:X\to Y$ is a constant.

Let $X=\mathbb R$ with the cofinite topology and $Y=\mathbb R$ with the usual topology. Show that any continuous map $f:X\to Y$ is a constant. My try: Let $f:X\to Y$ be continuous. If possible let ...
3
votes
2answers
70 views

If $A$ and $B$ are sets of real numbers, then $(A \cup B)^{\circ} \supseteq A^ {\circ}\cup B^{\circ}$

I have a proof for this question, but I want to check if I'm right and if I'm wrong, what I am missing. Definitions you need to know to answers this question: $\epsilon$-neighborhood, interior points ...
0
votes
1answer
53 views

Showing continuity of a function that depends on another continuous function.

Question: please help me pointing out the errors of my proof (I'm sure there are some). The proof is structured in cases (two cases with each two subcases) and I think that some may be correct but ...
1
vote
2answers
41 views

Finding the closure of $\{ \frac{1}{n} : n \in \mathbb{N} \}$ in finite complement topology on $\mathbb{R}$

Please point out where I am making a mistake: Let $\mathscr T$ be a finite complement topology of $\mathbb R$. Let K be defined as, $K= \{ \frac1n \mid n \in \mathbb N \}$. I need to find the ...
2
votes
3answers
110 views

Question about the boudary of a set $A \subseteq \mathbb{R}^n $.

let $A \subseteq \mathbb{R}^n$. Let $X = \{ x \in \mathbb{R}^n : \forall \epsilon > 0, \; \; B(x, \epsilon) \cap A \neq \varnothing \; \; and \; \; B(x, \epsilon) \cap ( \mathbb{R}^n \setminus A ) ...
3
votes
1answer
49 views

$\partial A = \overline{A} \cap \overline{ \mathbb{R}^d \setminus A } $

My attempt: ($A \subseteq \mathbb{R}^d$) We know by definition that $\overline{A} = \partial A \cup A $. Hence $\overline{ \mathbb{R}^d \setminus A} = \partial [ \mathbb{R}^d \setminus A ]\cup ...
0
votes
1answer
36 views

the boundary of set in euclidean space is closed.

MY Attempt: Let $\partial A$ be the boundary of any set $A \subseteq \mathbb{R}^n$. We show $R^n \setminus \partial A$ is open. Pick $x \in \mathbb{R}^n \subseteq \partial A $. Then by definition, we ...
2
votes
2answers
133 views

Every Cauchy sequence in a metric space $(X,d)$ is bounded.

MY attempt: Suppose $(x_n)$ is a Cauchy sequence in $(X,d)$. Take $\varepsilon = 1 $. Hence, can find $N$ such that $d(x_m,x_n) < 1 $ for all $n,m > N$. Also, we have $d(x_N, x_n) < 1 $ ...
1
vote
0answers
82 views

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
0
votes
0answers
45 views

Proof verification and suggestion to elude the AC (equivalent definition of adherent points).

Hi everyone I'd like to know if the following is correct and, more importantly, if there is some way to escape of the axiom of choice (as the hint the book says "use AC"). Definition: Let $X\subset ...