0
votes
2answers
28 views

Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
0
votes
1answer
52 views

A question about the proof of an obvious result

This is obviously true that a local homeomorphism is a continuous map. I tried to prove it this way : Suppose $f:X \to Y$ is a local homeomorphism, then $f$ is continuous if for each $x\in X$ and ...
1
vote
1answer
35 views

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$.

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$. I need to verify correctness of my proof and ask if there is a more straight-forward ...
4
votes
1answer
34 views

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$. Is $\cup \tau_\alpha$ a topology on $X$?

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$. Is $\cup \tau_\alpha$ a topology on $X$? For all $\alpha$, $\varnothing \in \tau_\alpha$ ...
2
votes
1answer
23 views

Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?

Can someone please verify my proof? Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$? No. Let $X = \mathbb{R}$. Clearly, $\{x\} \in \tau_\infty$ ...
1
vote
1answer
42 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
44 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
81 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
29 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
33 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
61 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
39 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
66 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
84 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
66 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
60 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
51 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
88 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
29 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
76 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
65 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 ) ...