Definition of Uniform Continuity in Arbitrary Topological Space. Let $(X,\mathfrak{T}_X)$ and $(Y,\mathfrak{T}_Y)$ be two topological space and let $f:X\to Y$. Then we say $f$ is uniformly continuous on $X$ if given an $\mathfrak{T}_Y$-open set $V$ of $Y$ with $f(x),f(y)\in V$, there exists an $\mathfrak{T}_X$-open set $U$ of $X$ such that $x,y\in U$ and $f(U)\subseteq V$.
I think that this definition of Uniform Continuity also implies Continuity and the proof of this immediately follows from the definition of Continuity,
Definition of Continuity. Let $(X,\mathfrak{T}_X)$ and $(Y,\mathfrak{T}_Y)$ be two topological space and let $f:X\to Y$. Then we say $f$ is continuous on $X$ if given an $\mathfrak{T}_Y$-open set $V$ of $Y$ with $f(x)\in V$, there exists an $\mathfrak{T}_X$-open set $U$ of $X$ such that $x\in U$ and $f(U)\subseteq V$.
So, I was wondering what problems will I face if I try to define Uniform Continuity in the manner I did above. Can anyone help?
Any continuous function $f:X\to Y$ is "uniformly continuous" according to your definition. Suppose $x,y\in X$ and let $V$ be any open set in $Y$ such that $f(x),f(y)\in V.$ Since $f$ is continuous, the set $U=f^{-1}(V)$ is an open set in $X$ with $x,y\in U$ and $f(U)\subseteq V.$
It's not just that this particular definition doesn't work. There is no way to define uniform continuity in terms of topologies, because various metrics can be compatible with one and the same topology; and a function can be uniformly continuous with respect to one metric and not uniformly continuous with respect to another metric, although both metrics induce the same topology.
However, you don't quite need a metric space to talk about uniform continuity. The most general setting for uniform continuity is uniform spaces.