0
votes
2answers
56 views

Suppose $x$ is a limit point of $A \subset X$, then if $f: A \to Y$ is continuous, is it true that $f(x)$ is a limit point of $f(A)$?

So I already know that a counterexample is $f(x) = c$ for $c$ is a constant, but I can't seem to prove this statement by contradiction, all I did was go back and forth. "Proof": If $f(x)$ ...
0
votes
1answer
43 views

Use contraction mapping theorem to prove

Help! I am taking a math course, and I just can't figure out this proof: Let $\alpha,\beta\in R^n$, $a\in R$, and $A$ be an $n\times n$ nonsingular matrix. Use contraction mapping theorem to prove ...
0
votes
0answers
46 views

Prove that a defined function g is continuous for a certain point

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Let $c \in \mathbb{R}$. Define the function $g_{c}: \mathbb{Q} \to \mathbb{Q}$ via: ...
2
votes
2answers
73 views

Prove that a function of the rational numbers $\mathbb{Q}$ with subspace topology inherited from $\mathbb{R}$ is injective

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Suppose $g: \mathbb{R} \to \mathbb{R}$ and $h: \mathbb{R} \to \mathbb{R}$ are ...
4
votes
3answers
42 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
1
vote
1answer
40 views

Proving properties of closures using intersection of indexed sets and topology

How would I write a proof for this example? Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an index set $I$ and $U_{i} \in B$ for each ...
0
votes
1answer
49 views

Question about proofs with topological spaces

How would I write a proof for this example? Let $(X, \tau_{1})$, $(Y, \tau_{2})$ and $(Z, \tau_{3})$ be topological spaces. A function ${f}: X \rightarrow Y$ is said to be continuous if for every V ...
0
votes
0answers
42 views

finite subset topology

Let $X$ be a set and $\tau=\left\{X\right\}\,\cup\,\{u_i\subset X|u_i \text{ is finite}\}$. Is $\tau$ is a topology on $X$? My effort to show this is as follows: 1) $X\in\tau$ by definition and ...
2
votes
3answers
69 views

Metric Space, Induced topology

I'm trying to show that the metric topology is indeed a topology. To do so, I want to show the following three statements are true: $\emptyset$ and X are in $\tau$ finite intersections of open sets ...
1
vote
2answers
75 views

Projections are open maps. Why might I be wrong?

I got this problem from Munkres, my idea is similar, but comparing to the actual solution, I missed at least 4 steps. Prove that the projection maps $\pi_1 : X \times Y \to X$ and $\pi_2 : X ...
3
votes
1answer
86 views

I want to figure out how many Topologies are in the set X

I have the set $$X = \{1, 2, 3\}$$ and I want to figure out how many different topologies I can get from the set $X$ so what I have done is assumed that the empty set and the whole set are in $T$ ...
2
votes
0answers
71 views

Set of limit points of S is closed in a metric space X

A point $x \in X$ is a limit point of a subset S of X, if every ball $B(x;\varepsilon)$ contains infinitely many points of S. Show that x is a limit point of S iff there is a sequence {$x_{j}$} ...
1
vote
1answer
42 views

If $S_n=\left[-\frac{n}{n+1},\frac{n}{n+1}\right]$ then $\bigcup\limits_{n\geq1}S_n=(-1,1)$

I was self reading Mathmatics for Economists by Simon and Blume. Consider the closed sets $S_n=\left[-\frac{n}{n+1},\frac{n}{n+1}\right]$ for $n\geq1,n\in\mathbb N$. Then ...
2
votes
2answers
105 views

Prove that the x-axis in R^2 with the Euclidean metric is closed

I want to show that the x-axis is closed. Below is my attempt - I would appreciate any tips on to improve my proof or corrections: Let (X,d) be a metric space with the usual metric. WTS: {(x,y) | X ∈ ...
2
votes
1answer
134 views

Proving infinite wedge sum of circles isn't first countable

Let $\{S_i\}_{i=1}^\infty$ be a countable family of circles and $\{p_i\}_{i=1}^\infty$ be a family of points such that $p_i\in S_i$. let $X = \bigcup _{i=1}^\infty S_i/\{p_i\}_{i=1}^\infty$ be the ...
2
votes
0answers
39 views

Is this proof correct: domain of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected.

The domain $X$ of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected. Proof: If $X = F \uplus G$ for two nonempty closed sets $F,G$ ...
3
votes
3answers
93 views

Is there a better way to write it?

I'm writting something. However I'm not good at English writting. Suppose that $X=D^\mathfrak c$. I want to express this : Let $x$ be the unique point of $X$ such that $x(\gamma)=1$ and for ...
0
votes
1answer
29 views

Is $X$ has a strong rank 1-diagonal?

Definition 1: A space $X$ has a strong rank 1-diagonal \cite{5} if there exists a sequence $\{\mathcal U_n: n\in \omega\}$ of open covers of $X$ such that for each $x\in X$, $\{x\}=\bigcap ...
0
votes
1answer
63 views

Could someone help me to improve the proof writing?

I will prove the following claim. I'm not a native English speaker. Could someone help me to improve the writing? A regular pseudocompact Moore space is ccc and first countable. Prove: I will ...
6
votes
2answers
2k views

Prove that if a set $E$ is closed iff it's complement $E^{c}$ is open

I was wondering if this proof was right. $\Leftarrow$ Suppose $E$ is closed. Then choose $x\in E^{c}$, then $x\notin E$, and so $x$ is not a limit point of $E$. Hence there exists a neighborhood ...
1
vote
1answer
52 views

Is this proof on the product of $X$ OK?

Let $X^2$ be star $\sigma$-compact and $F$ be a closed subset in $X^2$. If $\mathcal{U}$ is an open cover of $F$, then there exists a $\sigma$-compact subset $A$ of $X$, such that $F \subseteq ...
1
vote
1answer
49 views

Let $X$ has countable extent. Does $X^2$ have countable extent?

Definition 1: A space $X$ has countable extent if every uncountable subset of $X$ has a limit point in $X$. I'm struggling with this question: Question 2: Let $X$ has countable extent. Does ...
2
votes
1answer
44 views

Product of moscow spaces

Let $\{X_a : a\in A\}$ be a family of topological spaces such that $X_K=\prod\limits_{a\in K}X_a$ is a Moscow space of countable $o$-tightness, for every finite subset $K$ of $A$. Then the ...
5
votes
2answers
86 views

Product of a family of spaces of countable tightness

I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem? Finite family of compact spaces of countable ...
5
votes
1answer
78 views

Every first countable space is a moscow space.

First countable space $X$ is an example of moscow spaces. Let $U$ is an open subset of $X$ and $x\in \overline{U}$. If $\overline{U}$ is open or even a nbhood of $x$ this proposition is immediately ...
1
vote
2answers
121 views

Critique this proof on compactness.

Problem: Prove or disprove, the metric space $X$ containing infinitely many points with the discrete metric is compact. Write a proof in the language of sequences and covers Proof: Take $(1/n) \to ...
0
votes
2answers
275 views

Trying to prove noncompactness over the discrete metric

I wrote this version of proof. But would like some feedback. I am proving that the discrete metric $(X,d)$ is not compact. $X$ is infinite, it contains infinitely many points Let $x_i \in X$, then ...
8
votes
2answers
245 views

Why is the induction proof not sufficient? Topology…

I went to a exercise class and I got really confused. Consider the problem: Let $\lbrace A_{n} \rbrace$ be a sequence of connected subspaces of $X$, such that $A_{n}\cap A_{n+1}\neq \emptyset$ for ...
1
vote
2answers
310 views

Why is this last step even necessary in this proof with open sets?

Let $E^o$ denote the set of all interior point of a set $E$. Prove that $E^o$ is always open. Proof: For $p \in E^o$, there is a neighborhood $N_r (p) \subset E$. Since neighborhoods are open, for ...
1
vote
1answer
117 views

Proof on countably discrete subset of any Hausdorff space.

I have this question for several days: Let $X$ be a topological space and $X$ is Hausdorff. $C$ is an countable discrete subset of $X$. Then does there exist a disjoint family of open sets $\{U_x: ...
3
votes
1answer
69 views

How to show any convergent sequence is strongly discrete in Hausdorff space?

Given a space $X$ and $C \subset X$, we say that $C$ is strongly discrete if there exists a disjoint family $\{U_x: x\in C\} $ of open sets in $X$ such that $x\in U_x$ for all $x\in C$. The question ...
7
votes
3answers
3k views

A subset of a compact set is compact?

Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$. Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a ...
2
votes
0answers
3k views

Rudin 2.2: Prove the set of algebraic numbers is countable.

Similar to Proving that the set of algebraic numbers is countable without AC "A complex number $z$ is said to be algebraic if there are integers $a_0,\dots,a_n$, not all zero, such that $$ a_0 z^n + ...
0
votes
2answers
260 views

Proving a theorem from topology

Theorem: Suppose $Y \subset X$. A subset $E \subset Y$ is open relative to $Y$ if and only if $E = Y \cap G$ for some open subset G of X. I don't understand what's happening neither can I ...
8
votes
4answers
413 views

Should one imagine diagrams/figures when working?

I'm working through Baby Rudin and find it exceedingly difficult to understand what's happening without drawing a small figure. For instance when proving properties of compactness, I would often draw ...
4
votes
2answers
1k views

Prove that a set consisting of a sequence and its limit point is closed

Can someone please check whether the following simple proof is "mathematical"? Is it correct, complete, rigid? Can it be simplified? I'm a complete autodidact so I'm looking for someone to give me ...