# Questions tagged [selection-principles]

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7
questions

**24**

votes

**2**answers

1k views

### A set that can be covered by arbitrarily small intervals

Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of ...

**6**

votes

**3**answers

607 views

### When does the generalized Cantor space embed in a $\kappa$-compact space

The generalized Cantor space is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.
A space is $\kappa$-compact if ...

**7**

votes

**1**answer

272 views

### What is the height (or depth) of $[\mathbb{N}]^\infty$?

(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by $\...

**5**

votes

**1**answer

402 views

### When is there an unbounded tower in $[\mathbb{N}]^\infty$?

(Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.)
This question assumes familiarity with combinatorial cardinal ...

**4**

votes

**1**answer

434 views

### When is the generalized Cantor space $\kappa$-compact?

My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.
The generalized Cantor space is the space $2^\kappa$, with basic open ...

**18**

votes

**1**answer

696 views

### Two strengthenings of "strong measure zero"

A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering $X$...

**3**

votes

**1**answer

249 views

### Implications between different covering properties of spaces

Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a set,...