Compactness, continuity and the discrete topology Assume that $X, Y$ are compact metric spaces, and that there is a map
$$ \mu :  X \to \Delta (X \times Y)$$
such that $\mu$ is continuous, where $\Delta (\Omega)$ denotes the set of probability measures over a generic $\Omega$ . Endow $X \times Y$ with the product topology, and $ \Delta (X \times Y)$ with the topology of weak convergence.
The continuity of $\mu$ tells us that when we have an open (resp. closed) subset $G$ of $ \Delta (X \times Y)$, we are ensured that the preimage $\mu^{-1} (G)$ is open (resp. closed). However, I have a problem with the following situation.
Let $X$ be finite, hence compact. Let $G = \{ \delta_{(x,y)} \}$, where $\delta$ denotes the Dirac measure for some elements $x \in X$ and $y \in Y$. Thus $\mu^{-1} (\{ \delta_{(x,y)} \} )$  maps to some element $ x \in X$. But now, how can $\mu$ really be continuous in this case?  
Is it continuous because we are implicitly endowing the finite $X$ with the discrete topology?
Any feedback or answer is most welcome.
Thank you for your time.

PS: To the moderators, this questions looks fairly close to this previous one. However, they are different in spirit, because that question is not well written (too many questions into one). Hence, I decided to "unzip" it, starting from this one. Regarding this issue, I think it would be wise to close that linked question (I don't know how to do it).
 A: The constant map between two space is always continuous, in ANY topology you consider.
Suppose  $\mu$ is constant, that is to say $\mu(x_1)=\mu(x_2)$ for any $x_1,x_2\in X$ and suppose $\mu(x)$ is not a dirac delta.
Then $\mu^{-1}(\delta_{(x,y)})$ is just empty, so it is not an element of $X$.
This is not a particular case because if $X$ is finite, say with $n$ elements, then the image of $\mu$ consists of just $n$ measures, and the space of probability measures on $X\times Y$ contains infinitely many elements provided $n\geq 2$ and $Y\neq\emptyset$. 
Finally, if you endow $X$ with the discrete topology, then ANY map from $X$ to ANY topological space is continuous. On the other opposite, if $X$ has the trivial topology (the open sets are $X$ and $\emptyset$) then a function from $X$  to a $T_0$ space is continuous if and only if it is constant. Between these two cases you have intermediate cases. 
Example. Let $X=\{a,b,c\}$ with topology given by $\tau=\{X,\emptyset, \{a\}, \{b,c\}\}$ then a function from $X$ to a $T_0$ space is continuous if and only if $f(b)=f(c)$.
