If I have two topological spaces $(X,\mathcal{T}_x)$ and $(Y,\mathcal{T}_y)$. I am trying to show that the space $\{b\}\times Y$ is homeomorphic to Y, where $b\in X$
It is my understanding that if I find a function between the two spaces:
$$T: \{b\}\times Y\longrightarrow Y$$
that is continuous, bijective and $T^{-1}$ is also continuous then the two spaces are homeomorphic.
I figured that
$$T(b, y)=y$$
would suffice. I am self teaching myself topology. Is my function enough to justify that the two topological spaces are homeomorphic?
Best Answer
Yes, your $T$ is the restriction (to $\{b\} \times X_2$) of the second projection $\pi_2: X_1 \times X_2 \to X_2$, which is continuous by the definition of the product topology, and restrictions of continuous functions are still continuous.
Its inverse is $j: X_2 \to X_1 \times X_2$ defined by $j(x)=(b,x) \in \{b\} \times X_2$, which is continuous as $\pi_1 \circ j$ is constantly $b$ (so continuous) and $\pi_2 \circ j$ is the identity on $X_2$ (so continuous too), using the universal property of continuity of $X_1 \times X_2$. So $T$ is continuous with continuous inverse and so a homeomorphism.