1

I am studying the proof of theorem 5.5, regularity of the topology of separate continuity

We are familiar with Euclidean topology on $\mathbb{R}^2$, in which open sets are defined with respect to the $\epsilon$ balls centred at a point in the plane. Analogous to these $\epsilon$ balls, we define the '$\epsilon-$ plus' centred at a point $(a, b)$ as $$ P_\epsilon (a, b) = \{ (x, b) \in \mathbb{R}^2 : |x - a| < \epsilon \} \cup \{ (a,y) \in \mathbb{R}^2 : |y - b | < \epsilon \} .$$ We say $ U \subset \mathbb{R}^2$ is separately open set, if for each point in $U$, there is some $\epsilon > 0 $ such that $P_\epsilon (a,b) \subset U$. These separately open sets form a topology on $\mathbb{R}^2$, called separately open topology. It is denoted as $\mathbb{R \otimes R}$. when we restrict it to $\mathbb{Q}^2$ we get the topology $\mathbb{Q \otimes Q}.$ In particular, I want to show that $\mathbb{Q \otimes Q}$ is regular.

Now, in the proof of theorem 5.5; i do not understand the notation and what they mean by $(\gamma)$ and $(\delta)$.

In the construction given in second paragraph, I have following doubts:

  1. $E_\epsilon$ cover the closed set A. and since A is closed we can find the countable number of closed $A_j \subset A$, such that $A = \cup_j A_j$. but in the construction above A is a Clopen set. please explain it.

  2. how is the $diam(R_j) < 1/j$ can satisfy $(R_j \times A_j) \cap K_{n-1} = \phi$? enter image description here

Ashutosh Shinde
  • 31
  • The space $X\times Y$ and the notations $E_x$, $E^y$ were defined in Definition 1.1 of this paper. $S^\circ$ denotes the interior of $S.$ This answers your first question "i do not understand understand the notation and what they mean by (γ) and (δ); Please explain the meaning". As for the second one "and further, can you explain how this construction can be done?", it needs more focus, since this construction is detailed in the (very long) second paragraph of their proof: on which step precisely are you stuck? – Anne Bauval Nov 20 '23 at 18:09

1 Answers1

1

The space $X\otimes Y$ and the notations $E_x,E^y$ were defined in Definition 1.1 of this paper. $S^∘$ denotes the interior of $S.$ This answers your first question "i do not understand the notation and what they mean by $(γ)$ and $(δ)$".

As for your (twofold) second question:

  1. The partition of $Y$ into two clopen subsets $A,B$ such that $(H_{n-1})_x\subseteq A$ and $(K_{n-1})_x\subseteq B$ is prior to the definition of the $E_\varepsilon$'s. The existence of such a partition is due to the fact that $(H_{n-1})_x,(K_{n-1})_x$ are closed and disjoint (since $H_{n-1},K_{n-1}$ are, by induction hypothesis), and $Y$ is strongly $0$-dimensional.
  2. Since $\inf\{d(x,(K_{n-1})^y)\mid y\in A_j\}>0,$ for every $x'$ sufficiently close to $x$ and every $y\in A_j$, we have $(x',y)\notin K_{n-1}.$ Now, $x$ has a basis of clopen neighborhoods, so this "$x'$ sufficiently close to $x$" can be replaced by $x'\in R_j$, for some "small" (e.g. with diameter $<1/j$) clopen neighborhood $R_j$ of $x$.
Anne Bauval
  • 34,650
  • Thank you for the answer. I understood the (2)part. But my question (1.) is "A ,B are a clopen sets, But in paraghraph 2 - 7th line, they said A is a closed set. How is this possible? Please help. – Ashutosh Shinde Nov 22 '23 at 07:37
  • $A$ was chosen clopen hence it is closed. – Anne Bauval Nov 22 '23 at 08:08
  • In (1.), can you please give any hints to proceed or show the existance of partition using the definition of strongly 0 dimenational space. i.e. A topological space X is called strongly zero-dimensional if X is a non-empty Tychonoff space and every finite functionally open cover ${U_i}{i=1}^{k}$ of the space $X$ has a finite open refinement ${V_i}{i=1}^{m}$ such that $V_i \cap V_j = \phi$ whenever i$ \not = j$. – Ashutosh Shinde Nov 22 '23 at 15:46
  • 1
    The authors of your paper took the existence of such a partition $(A,B)$ for every two disjoint closed subsets $H,K$ (or equivalently: the existence of a base of clopen subsets) as a definition of strong $0$-dimensionality. They used it in their proof of Theorem 2.12.b. I am not an expert on this subject. For various equivalent definitions, see this discussion and references. In particular Engelking Theorem 6.2.4 seems to be what you are asking for. – Anne Bauval Nov 22 '23 at 16:44
  • One more question: In the first paragraph how does the $\beta, \gamma , \delta$ imply the each $U_x$ and each $U^y$ is clopen? – Ashutosh Shinde Nov 23 '23 at 01:59
  • Use that $S^∘$ denotes the interior of $S.$ If you have new questions, please edit new posts. – Anne Bauval Nov 23 '23 at 05:50