# Doubt in homeomorphism

A mapping is a homeomorphism if it is a continuous, bijective map such that it's inverse is also continuous. This is the definition of homeomorphism which was taught to me. What does the following statement mean:"A point P is homeomorphic to an open disc in $$\mathbb{R}^2$$".

• It's a false assertion. Do you have more context? Feb 22, 2021 at 16:38
• In that definition you were taught, a homeomorphism is a kind of mapping. The concept of homeomorphic is closely related, but not exactly the same. It is a relation amongst topological spaces: given two topological spaces $X$ and $Y$, to say that $X$ and $Y$ are homeomorphic means that there exists a homeomorphism $f : X \to Y$. As the answer of @AlekosRobotis explains, that statement in your post is false because no homeomorphism exists between those two spaces. Feb 22, 2021 at 16:50

The statement says that there exists a continuous bijection from $$\{p\}\to D\subseteq \Bbb{R}^2$$ where $$D$$ is an open disk in $$\Bbb{R}^2$$ with continuous inverse. That is, there exists a homeomorphism $$\phi:\{p\}\to D$$. This is, of course, false. There is not even a bijection between $$\{p\}$$ and $$D$$. One of them has cardinality $$1$$, while the other has uncountable cardinality.
More generally, let $$(X,\mathscr{T}_X)$$ and $$(Y,\mathscr{T}_Y)$$ be topological spaces. We say that $$(X,\mathscr{T}_X)$$ and $$(Y,\mathscr{T}_Y)$$ are homeomorphic if there exists a homeomorphism $$X\to Y$$; that is, a bijection $$\phi:X\to Y$$ which is continuous (with respect to the topologies $$\mathscr{T}_1$$ and $$\mathscr{T}_2$$) and which has a continuous inverse $$\phi^{-1}$$. We often shorten this and say that $$X$$ and $$Y$$ are homeomorphic if the topologies are clear, and we often write $$X\cong Y$$ (or similar notations).
In your case, we are abbreviating the statement that $$\big(\{x\},\mathscr{T}^\text{ind}_{\mathbb{R}^2}\big)\cong\big(\mathbb{R}^2,\mathscr{T}_{\mathbb{R}^2}\big)$$ (already demonstrated by other answers to be false) as $$\{x\}\cong\mathbb{R}^2$$, where $$\mathscr{T}_{\mathbb{R}^2}$$ is the Euclidean topology on $$\mathbb{R}^2$$ and $$\mathscr{T}^\text{ind}_{\mathbb{R}^2}$$ is the induced topology on the point set $$\{x\}\subseteq\mathbb{R}^2$$.