I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R}$} as it's base.

If $X$ is a topological space, an upper semi-continuous real function on $X$ can be interpret as a continuous function from $X$ into $\mathbb{R}_l$.

The set of all upper semi-continuous real functions on $X$ is denoted by $USC(X)$.

We could easily see that if we consider $X=\mathbb{R}$ then the set $USC(\mathbb{R})$ is not a ring with pointwise addition and multiplication. because there is $f \in USC(\mathbb{R})$ so that $-f \notin USC(\mathbb{R})$. Indeed it is not a Group.

Question.For what condition(s) on $X$, the set $USC(X)$ constructs a ring structure we the pointwise addition and multiplication?

PS:I am looking for the topological condition(s) $P$ on $X$ so that, $USC(X)$ is a ring iff $X$ has the property $P$.

Thank you so much for noticing to my Question.

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# The set of Upper semi-continuous functions as a ring.

I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R}$} as it's base.

If $X$ is a topological space, an upper semi-continuous real function on $X$ can be interpret as a continuous function from $X$ into $\mathbb{R}_l$.

The set of all upper semi-continuous real functions on $X$ is denoted by $USC(X)$.

We could easily see that if we consider $X=\mathbb{R}$ then the set $USC(\mathbb{R})$ is not a ring with pointwise addition and multiplication. because there is $f \in USC(\mathbb{R})$ so that $-f \notin USC(\mathbb{R})$. Indeed it is not a Group.

Question.For what condition(s) on $X$, the set $USC(X)$ constructs a ring structure we the pointwise addition and multiplication?