When we study Hausdorff space, we can have the following two results:
Cor. 173
If $f,g: X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\mid f(x) = g(x)\}$ is closed in $X$.
Cor. 174
If $f,g:X \to Y$ are continuous and $Y$ is $T_2$ and $\{x \in X\mid f(x)=g(x)\}$ is a dense subset of $X$, then $f=g$.
In fact, these two statements can be used to characterize Hausdorff.
For 173, we have characterization 1:
if $Y$ is not Hausdorff, there always exists a topological space $X$ and two continuous functions $f$ and $g$, such that the set $\{x | f(x) = g(x)\}$ is not closed.
Thus, “a space Y is Hausdorff if and only if for every topological space X and for any continuous maps from X to Y, the set $\{x | f(x) = g(x)\}$ is closed.” is a characterization of Hausdorff Space.
174 is similar, we have characterization 2:
$Y$ is Hausdorff, if and only if whenever we have $f$ and $g$ continuous from any space $X$, and $\{x | f(x) = g(x)\}$ is a dense subset of $X$, then $f=g$”
My question is: How to prove these, i.e., How to find examples to prove the other direction of characterization 1,2?
Characterization 2 already has an answer here: Does this property characterize a space as Hausdorff?
For any directed set $I$ (such as we use in nets) we define $X(I):=I \cup \{\infty\}$ where $I$ consists of isolated points and a basic neighbourhood of $\infty$ is of the form $\{\infty\} \cup \{i \in I: \mid i \ge i_0\}$ where $i_0 \in I$ is arbitrary. It's easily shown that this defines a topological space. In it$ I$ is dense and non-closed.
Then if $Y$ is not Hausdorff we have a net $(x_i)_{i \in I}$ in $Y$ that converges to two distinct points $p\neq q \in Y$.
Then defining $f,g: X(I) \to Y$ by $f(i)=x_i = g(i)$ and $f(\infty)=p, g(\infty)=q$ we easily check that both are continuous maps (due to the net converging to both points), and $I=\{x\mid f(x)=g(x)\}$ is both not a closed set and also dense despite $f \neq g$. So both claims are shown at the same time.