A topological space $A$ that is also a ring with operations '+' and '.' where $ A\times A$ carries product topology, is a semi-topological ring if the mappings $$a_1: A\times A \to A\ \text{such that}\ (x,y)\to x+y$$ , $$a_2: A \to A\ \text{such that}\ x\to -x$$ and $$a_3: A \to A\ \text{such that}\ (x,y) \to x.y$$ are semi-continuous.

A topological space $A$ that is also a ring with operations '+' and '.' where $ A\times A$ carries product topology, is a B-topological ring if the mappings $$a_1: A\times A \to A\ \text{such that}\ (x,y)\to x+y$$ , $$a_2: A \to A\ \text{such that}\ x\to -x$$ and $$a_3: A \to A\ \text{such that}\ (x,y) \to x.y$$ are b-continuous.

where,

A function $f:X \to Y$ from topological space $X$ to topological space $Y$ is said to be b-continuous (semi continuous) if for each $x\in X$ and each open set $U \subset Y$ containing $f(x)$ there exists a b-open (semi-open) set $V \subset X$ containing $x$ such that $f(V) \subset U$.

A set $B$ is called b-open if $B \subset Int(cl(B)) \cup cl(Int(B))$ and $S$ is semi open if $S \subset cl(Int(S))$

Clearly, Topological rings $\implies$ semi-topological rings $\implies$ B-topological rings but not conversely.

After all these definitions I seek a B-topological ring that is not semi-topological. I already have other counterexamples but this one I have no idea about.

A topological space $A$ that is also a ring with operations ‘$+$’ and ‘$.$’, is a semi-topological ring if the mappings \begin{gather*} a_1: A\times A \to A\text{ such that }(x,y)\to x+y, \\ a_2: A \to A\text{ such that }x\to -x \end{gather*} and $$a_3: A \to A\text{ such that }(x,y) \to x.y$$ are semi-continuous where $ A\times A$ carries product topology.

A topological space $A$ that is also a ring with operations ‘$+$’ and ‘$.$’, is a B-topological ring if the mappings \begin{gather*} a_1: A\times A \to A\text{ such that }(x,y)\to x+y, \\ a_2: A \to A\text{ such that }x\to -x \end{gather*} and $$a_3: A \to A\text{ such that }(x,y) \to x.y$$ are b-continuous where $ A\times A$ carries product topology.

Where,

A function $f:X \to Y$ from topological space $X$ to topological space $Y$ is said to be b-continuous (semi-continuous) if for each $x\in X$ and each open set $U \subset Y$ containing $f(x)$ there exists a b-open (semi-open) set $V \subset X$ containing $x$ such that $f(V) \subset U$.

$\DeclareMathOperator\Int{Int}\DeclareMathOperator\cl{cl}$A set $B$ is called b-open if $B \subset \Int(\cl(B)) \cup \cl(\Int(B))$ and $S$ is semi-open if $S \subset \cl(\Int(S))$.

Clearly, topological rings $\implies$ semi-topological rings $\implies$ B-topological rings but not conversely.

After all these definitions I seek a B-topological ring that is not semi-topological. I already have other counterexamples but this one I have no idea about.

A topological space $A$ that is also a ring with operations '+' and '.' where $ A\times A$ carries product topology, is a semi-topological ring if the mappings $$a_1: A\times A \to A\ \text{such that}\ (x,y)\to x+y$$ , $$a_2: A \to A\ \text{such that}\ x\to -x$$ and $$a_3: A \to A\ \text{such that}\ (x,y) \to x.y$$ are semi-continuous.

A topological space $A$ that is also a ring with operations '+' and '.' where $ A\times A$ carries product topology, is a B-topological ring if the mappings $$a_1: A\times A \to A\ \text{such that}\ (x,y)\to x+y$$ , $$a_2: A \to A\ \text{such that}\ x\to -x$$ and $$a_3: A \to A\ \text{such that}\ (x,y) \to x.y$$ are b-continuous.

where,

A function $f:X \to Y$ from topological space $X$ to topological space $Y$ is said to be b-continuous (semi continuous) if for each $x\in X$ and each open set $U \subset Y$ containing $f(x)$ there exists a b-open (semi-open) set $V \subset X$ containing $x$ such that $f(V) \subset U$.

A set $B$ is called b-open if $B \subset Int(cl(B)) \cup cl(Int(B))$ and $S$ is semi continuous if $S \subset cl(Int(S))$

Clearly, Topological rings $\implies$ semi-topological rings $\implies$ B-topological rings but not conversely.

After all these definitions I seek a B-topological ring that is not semi-topological. I already have other counterexamples but this one I have no idea about.

A topological space $A$ that is also a ring with operations '+' and '.' where $ A\times A$ carries product topology, is a semi-topological ring if the mappings $$a_1: A\times A \to A\ \text{such that}\ (x,y)\to x+y$$ , $$a_2: A \to A\ \text{such that}\ x\to -x$$ and $$a_3: A \to A\ \text{such that}\ (x,y) \to x.y$$ are semi-continuous.

A topological space $A$ that is also a ring with operations '+' and '.' where $ A\times A$ carries product topology, is a B-topological ring if the mappings $$a_1: A\times A \to A\ \text{such that}\ (x,y)\to x+y$$ , $$a_2: A \to A\ \text{such that}\ x\to -x$$ and $$a_3: A \to A\ \text{such that}\ (x,y) \to x.y$$ are b-continuous.

where,

A function $f:X \to Y$ from topological space $X$ to topological space $Y$ is said to be b-continuous (semi continuous) if for each $x\in X$ and each open set $U \subset Y$ containing $f(x)$ there exists a b-open (semi-open) set $V \subset X$ containing $x$ such that $f(V) \subset U$.

A set $B$ is called b-open if $B \subset Int(cl(B)) \cup cl(Int(B))$ and $S$ is semi open if $S \subset cl(Int(S))$

Clearly, Topological rings $\implies$ semi-topological rings $\implies$ B-topological rings but not conversely.

After all these definitions I seek a B-topological ring that is not semi-topological. I already have other counterexamples but this one I have no idea about.