# Tagged Questions

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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 $V=X−U$, 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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### Metric-like families of relations

Let $X$ be an arbitrary set and to start with, let us consider a relation $\leq$ on $X$ (that is $\leq$ is a subset of $X^2$) which is reflexive and transitive. such a relation is called a preorder. ...
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### If a topological space $X$ has a countable basis. Then if we have an open cover of $X$, can this cover be refined to a countable one?

If a topological space $X$ has a countable basis. Then if we have an open cover of $X$, can this cover be refined to a countable one?
Count the number of topological sorts for each partially ordered set $(A,|)$, where (a) $A = (3, 5, 7, 11, 13, 16, 17)$ (b) $A = (1, 3, 9, 27, 81, 243)$ That is, you have to ﬁnd the number of ways ...