In this image , There are some definitions of toplogical groups . But I am confused to understand , How he check the continuity of binary operations. Please explain it by giving suitable examples . Here Nx is collection of neighbourhoods of x . Thanks in advance .
A simple example:
Suppose we have $s: \Bbb R^2 \to \Bbb R$ given by $s(x,y) = x+y$, with the usual metric topology on $\Bbb R$.
Here, $N(x_0)_{\delta} = \{x \in \Bbb R: |x-x_0| < \delta\}$, and similarly for $N(y_0)_{\delta}$, and the equivalent formulation of Proposition 3.4 is that given:
$N(x_0+y_0)_{\epsilon} = \{t \in \Bbb R: |t - x_0 - y_0| < \epsilon\}$ we can find $\delta_1,\delta_2$ such that:
$N(x_0)_{\delta_1} + N(y_0)_{\delta_2} \subseteq N(x_0+y_0)_{\epsilon}$.
If we choose $\delta_1 = \delta_2 = \dfrac{\epsilon}{2}$, we have:
$|x + y - (x_0 + y_0)| = |(x - x_0) + (y - y_0)| \leq |x - x_0| + |y - y_0| < \dfrac{\epsilon}{2} + \dfrac{\epsilon}{2} = \epsilon$, that is to say:
If $x \in N(x_0)_{\frac{\epsilon}{2}}$, and $y \in N(y_0)_{\frac{\epsilon}{2}}$, that is $x + y \in N(x_0)_{\frac{\epsilon}{2}} + N(y_0)_{\frac{\epsilon}{2}}$,
then $x + y \in N(x_0+y_0)_{\epsilon}$, which is simply another way of saying:
$N(x_0)_{\frac{\epsilon}{2}} + N(y_0)_{\frac{\epsilon}{2}} \subseteq N(x_0+y_0)_{\epsilon}$.
(finding suitable neighborhoods of $x_0,y_0$ is particularly easy in this case, since $s$ is linear in $x$ and $y$, because of the distributive property of multiplication in fields).
The function $\iota: \Bbb R \to \Bbb R$ given by $\iota(x) = -x$ is also easily shown to be continuous in this topology (we can take $\delta = \epsilon$), and thus $(\Bbb R, +)$ with the $\epsilon$-interval neighborhood system is a topological group.
Caution: proving a particular group operation in a particular topological space is continuous may entail a bit more work than this particular example. But the usual topological groups studied (such as $GL_n(\Bbb R)$, or $(\Bbb R^n,+)$ or $S^1 = \{z \in \Bbb C: |z| = 1\}$ under complex-multiplication) arise from operations well-known to be continuous. For finite groups, for example, one is often forced to use the discrete topology, as no other will work.