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In the theorem 3.6 of Juhász's Cardinal Functions in General Topology appears the following symbol about sequence: $\frown$

The role context of it's appearance is the following:

Theorem. Let X be an compact topological space and $\kappa$ an cardinal such that $\chi(p,X)\geq \kappa$, for all $p\in X$. Then $|X|\geq 2^\kappa$.

Proof. First, set $\kappa =\omega$; we shall prove a little more than stated, namely that $X$ can be mapped continuously onto the interval $[0,1]$. To achieve this, we first define be induction on $n\in\omega$ an non-empty open subset $U_{\varepsilon}$ of $X$ for each finite sequence $\varepsilon\in 2^n$ in such way that

  1. $\overline{U_{\overset{\frown}{\varepsilon 0}}} \cup \overline{U_{\overset{\frown}{\varepsilon 1}}} \subseteq U_{\varepsilon}$
  2. $\overline{U_{\overset{\frown}{\varepsilon 0}}} \cap \overline{U_{\overset{\frown}{\varepsilon 1}}} = \emptyset$

The proof goes on...

What is the name of this symbol and what does it means?

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2 Answers 2

up vote 2 down vote accepted

Usually this means extending the string $\varepsilon$ by appending $x$ to it at the end, or concatenating two strings.

So $(0,0,1)^\frown 0=(0,0,1,0)$.

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I suppose it means concatenation of sequence $\varepsilon$ with the one element $0$ or $1$.

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