I'm studying about manifolds. I'm an engineer and want to grasp the concept of manifold without too deep dive into rigorous mathematics. My question is whethere the "topologist's sine curve" defined as $S = \{(0, y) : y \in (-1,1)\} \cup \{(x, \sin (1/x)) : x \in (0, 2/\pi)\}$ can be a manifold. I tried to find some explnations in books and web. However, what I found are all about (path/local) connectedness. I'm not familiar with topology and not want to study it extensively. So I tried to understand the question with very minimal knowledge.
As far as I understand, to say that a set is a manifold, I need to set a topology (open subsets) on the set. As a subset of $\mathbb{R}^2$, every open set in $S$ should have a form $S \cap A$ for an open set $A \in \mathbb{R}^2$. Any interval in the vertical axis part of $S$, e.g. $I = \{(0, y) : 0 < y < 1/2\}$, cannot be an open set in $S$. It is because every open set in $\mathbb{R}^2$ containing $I$ also contains some other part of $S$ : there is no way to express $I$ in the form of $S \cap A$ where $A$ is open in $\mathbb{R}^2$. Then there is no homeomorphism from an open subset of $S$ containing $I$ to an open set in $\mathbb{R}$. It is because, if there is one, the preimage of an open interval in $\mathbb{R}$, to which $I$ is mapped, must be open but $I$ is not open. So there cannot be a coordinate chart for $I$ and $S$ is not a manifold. This is what I understand. My first question is whether or not this understanding is correct.
The second question is whether $S$ can be a manifold if I give a different topology. I cannot express precisly how to define the topology but intuitively the sine function part is homeomorphic to an open interval in the real line and the vertical part is also homeomorphic to an open interval in the real line, if I consider $S$ as a space of its own, not a subspace of $\mathbb{R}^2$. Is this true? If true, how can I express the topology precisley?
This question is not a duplicate of Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?.
I read it. However, the concept of connectedness confuses me. As I said in the first paragraph, I want to understand my question with minimal knowledge on topology such as just the definition of open sets, continuous functions and manifolds. Moreover the linked question does not concern my second question.
Best Answer
The topologist's sine curve $S$ is a subspace of $\mathbb{R}^2$ meaning that it is a subset of $\mathbb{R}^2$ and inherits its topology from the topology of $\mathbb{R}^2$. In order to be a manifold it must be locally Euclidean in the inherited topology. That means that it must also be locally connected but, as noted, it is not locally connected on $0\times[-1,1]$.
You ask whether perhaps some other topology on $S$ would make it a manifold.
Perhaps, but with some topology other than the one it inherits from $\mathbb{R}^2$ it is no longer the topologist's sine curve.