What are counter examples for these statements?

Question

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...

Answer 1

For question one take $I=\varnothing$.

For question two take $X=Y=\{0,1\}$ with the discrete resp. trivial topologies, i.e. $$T_X=\{\varnothing,\{0\},\{1\},\{0,1\}\}\qquad\text{ and }\qquad T_Y=\{\varnothing,\{0,1\}\}.$$ Then taking for $f:\ X\ \longrightarrow\ Y$ the identity will do.

Answer 2

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.