# Tagged Questions

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### Properties of preimages and intersections of sets

I am working through Bert Mendelson's "Introduction to Topology" and am having some trouble with proofs. The text in well presented but to get a proper understanding I am working through the ...
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Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
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### Sum of Neighborhoods of Zero

When do two neighborhoods of zero over a topological vector space add up as: $$aN+bN=(a+b)N\quad a,b\geq 0$$ I could imagine something like balanced might suffice... The problem is that I'd like to ...
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### Regarding open subsets in topology

(Probably due to my lack of experience with the subject, I see that my question is horribly written. If you are to answer, a beginner-friendly explanation of the basis of a topology and the topology ...
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### Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?

I understand that when we talk about union of open sets, we introduce an index set which can be countable or uncountable. But could I not do the same for the intersection of open sets too?
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### Question about proof on basis

I found this proof online, but I have a bit of trouble understanding it. Question: Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an ...
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### The union of all the open sets in a family of topologies

I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let $\{\tau_\alpha\}$ be a family of topologies on X. [...] To say that ...
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### Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$A_x :=\{y\in Y:(x,y)\in A\}$$ and similarly for $\bar A$. Consider a ...
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### How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets? (or equivalently, that $\mathbb R$ contains no more then $\mathfrak c$ $G_\sigma$ sets? The more general ...
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### Why it is always circle to represent a Set?

When we draw a Venn diagram, we use circle to represent a Set. We can use any closed plane figure but most of the time it is a circle. Why? are there any specialty about that?
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### Calculate the number $o(\mathbb{R})$ of open subsets of the real line. [duplicate]

Calculate the number $o(\mathbb{R})$ of open subsets of the real line. I know that the answer is $\mathfrak{c}$ but I don't know how my lecturer got this. I am doing an introductory topology course, ...
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### Basic topology questions with cantor's set

I have 3 questions in toplogy, one of which I managed to solve (but would appreciate input regardless) and 2 which are more difficult. I'd like a push in the right direction. Define $K$ as ternary ...
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### Hausdorff topologies on the natural number set are sigma algebra

Is it true that if I add the Hausdorffness condition to any topology on $\mathbb{N}$, then it is a $\sigma$- algebra on $\mathbb{N}$? Once I have tried to prove this, I think that compactness is also ...
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### Denseness of a set, whose complement is known to be dense.

Do you think in general that if say $U\subset X$ was dense in $X$, then if we let $V=Xâˆ’U$, but we know $V$ is of higher cardinality than $U$, does that imply that $V$ must be dense?
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### How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
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### Are the axioms of a topological space superfluous?

A topology on a set $X$ is a family $\mathcal{T}$ of subsets of $X$, which are open sets and satisfy: (1) $\emptyset, X \in \mathcal{T}$. (2) Any union of elements of $\mathcal{T}$ belongs to ...
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### Proof that a given projection map restricted to a subset is closed.

$\pi_{1}:\mathbb{R}^2\rightarrow\mathbb{R}, (x,y)\mapsto x$ is a projection map from $\mathbb{R}^2$ with the standard eulcidean topology, $\mathscr{T}_E$ to $\mathbb{R}$ with it's usual euclidean ...
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Homework problem, intro to topology. Here's what I've done so far. Am I on the right track? And, how would you advise me to proceed from here? I have already established that $\left |[0,1] \right | = ... 1answer 41 views ### Problem involving an infinite lattice grid I'm stuck on this problem for Intro to Point-Set Topology.... I'm given that a submarine starts somewhere in$\mathbb{R}^2$and moves in a straight line at constant velocity, in such a way that at ... 3answers 55 views ### does the domain can be considered as subset of it image under 1 to 1 function? Let$f\colon X \to X$be a one-to-one function and let$A \subseteq X$. Does$A \subseteq f(A)$? I ask because I found a step which not clear to me in this paper ... 4answers 436 views ### Do these theorems about power sets hold for the empty set? From the definition of the power set as the set of all subsets of a given set, I realize that$\mathcal P(\varnothing) = \{ \varnothing \}$, in other words, the power set of the empty set is the set ... 2answers 68 views ### The difference between a finite set and an ordered$n-\$tuple? Proving the set of all finite subsets of a countable set is countable.

For my point-set topology class, I'm working on proving the theorem: The set of all finite subsets of a countable set is countable. Please don't post the proof of the theorem. The proof was easy for ...