# Tagged Questions

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### Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
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### What's the diference between $A<\infty$ and $A<\aleph_0$?

In my topology class the teacher gave some examples of topologies, and I'm trying to prove that they really are topologies. If $X$ is a set then: $\mathcal C=\{A:\# (X-A)<\infty\}$ is a topology ...
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### Is this sentence OK?

I'm starting to write a paper. This is the sentence which I want to put first in the paper. It is well known that diagonal properties are useful in estimating certain cardinal invariants of a ...
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### cardinality of the set of all dense subsets of $\Bbb R$

Let $$A=\{X \subseteq \mathbb R : \operatorname{cl}(X)=\mathbb R\}$$ Prove that the set $A$ and $P(\mathbb R)$ have the same cardinality. Well, the first thing it came to my mind was the injective ...
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### Cardinality of the class of $G_\delta$ subset of $\mathbb{R}$ of Lebesgue measure zero

Let $\mathcal{N}$ be the class of all subsets of $\mathbb{R}$ of Lebesgue measure zero and let $\mathcal{G}_\delta$ be the class of all $G_\delta$ subsets of $\mathbb{R}$. How do I show that ...
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### If $X$ is Hausdorff and $|X|> \mathfrak{c}$, does $X$ always have a uncountable discrete subspace?

Let $X$ be a Hausdorff topological space with $|X|> \mathfrak c$. Does $X$ always have a uncountable discrete subspace? Thanks for your help.
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### Tightness and countable intersection of neighborhoods

The following is a problem a colleague has encountered. He would like to know whether the following conjecture is right, wrong, or neither: Let $X$ be a topological space of countable tightness ...
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### $\kappa\psi (x,X)\leq \psi (x,X)$

The $\kappa$-pseudocharacter $\kappa\psi (x,X)$ of a space $X$ at a point $x\in X$ is the smallest infinite cardinal number $\tau$ such that there exists a family $\gamma$ of $\kappa$-sets in $X$ ...
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### 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 ...
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### “big” Hausdorff space with dense subspace of given cardinality

In a topology course we proved the following proposition: Let $A$ be an infinite set. Then there exists a Hausdorff space $X$ of cardinality $|\mathfrak{P}(\mathfrak{P}(A))|$ which contains a ...
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### How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
CCC means countable chain condition; A cover $\cal A$ of a set $E$ is separating if for each $p\in E$, $\bigcap \{A: A \in \mathcal{A}, p\in A\}=\{p\}.$ The point separating weight of $X$, denoted ...