What are counter examples for these statements? 
Question 1. 
Let $\{T_i\}_{i\in I}$ be a family topologies on a set $X$.
Provide an example that $\bigcup T_i$ is not a topology on X.

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Question 2.
Let $X$ be a compact space and $Y$ be a topological space.
Let $f:X\rightarrow Y$ be a continuous bijection.
Provide an example that $f$ is not a homeomorphism.

What are these two counter examples?
For Question 1, I think two non-comparable(for the order $\subset$) topologies should be given, but I cannot think of one since every finite dimensional normed space has the unique topology under any norm. I cannot think of an example that is not a finite-dimensional topological space.
For Question 2, $Y$ must not be Hausdorff. Likewise above, I cannot think of an explicit example which is not Hausdorff...

A non-trivial example for question one: let $X = \{1,2,3\}$.
Take $T_1 = \{\emptyset,\{1\},X\}$, and $T_2 = \{\emptyset,\{2\},X\}$.  Note that $T_1 \cup T_2$ is not a topology.
