Definition of a self homeomorphism

Question

In this article https://en.wikipedia.org/wiki/Homeomorphism

The definition of a homeomorphism is stated as follows:

A function $f : X \rightarrow Y$ between two topological spaces $(X,T_X)$ and $(Y, T_Y)$ is called a ''homeomorphism'' if it has the following properties:

• $f$ is a bijection,

• $f$ is continuous,

• the inverse function $f^{-1}$ is continuous.

Now what if we have the same space $X$ but with two given topologies $T_X$ and $T^{'}_X$ can we say from the definition that $(X,T_X)$ and $(X,T^{'}_X)$ are homeomorphic if and only if $T_X=T^{'}_X$ ?

Answer 1

No; consider the set $X=\{0,1\}$ with the topologies $$T_X=\{\varnothing,\{1\},X\} \qquad\text{ and }\qquad T_X'=\{\varnothing,\{0\},X\}.$$ Then $T_X\neq T_X'$ but the bijection $$(X,T_X)\ \longmapsto\ (X,T_X'):\ x\ \mapsto\ 1-x,$$ is a homeomorphism.

Answer 2

Not at all. One example is the right half-open interval and left half-open interval topologies on $\mathbb{R}$ with $f(x) = -x$.

Suppose $\mathscr{T}$ is a topology on $X$ and $f : X \rightarrow X$ is a bijection which is not continuous if both have that topology. Then $(X,f(\mathscr{T}))$ is homeomorphic to $(X,\mathscr{T})$ but not $\mathscr{T}$ and $f(\mathscr{T})$ are not equal.