I found the following question in a GRE prep resource, and I don't understand what the question is asking. In particular, I don't know what the ~ symbol means in analysis and topology, and my attempts to look this up elsewhere kept giving me results from statistics.
The question is: "Let $X$ be $\mathbb{R}$ with the topology given by letting the cocountable sets be open. Let $Y = (X \times [0, 1]) / ((x, t) \sim (x', t') \Longleftrightarrow t = t' = 1)$. Which of the following is false?"
I can also include the answer options here if people want, but I don't think that's relevant. I'm just looking for clarification about what this question is asking and what the tilde means.
In this case, the symbol represents an identification. The notation $$Y = (X\times[0,1])/((x,t)\sim(x',t') \iff t = t' = 1) $$ is defining a new space $Y$. This space $Y$ starts out as the Cartesian product of some other topological space $X$ and the closed unit interval $[0,1]$, endowed with the natural product topology. For example, if $X$ is a unit square, $X \times [0,1]$ is a unit cube. In the problem, $X = \mathbb{R}$, so $X \times [0,1]$ is an infinitely long "band" that is one unit wide. Do note, however, that the real numbers have been endowed with a non-standard topology, so the topology on this band will also not be the "usual" topology inherited from $\mathbb{R}^2$.
So $Y$ starts as this infinite band. But it is further modified by "modding out" by an equivalence relation. What this means is that certain points are being declared as identical, and "glued" together. In this case, two points are "the same" if their $y$-coordinates are the same. Essentially, all of the points of the form $(x,1)$ are being smooshed down into a single point.
The resulting object is a "cone" whose base is the real line.
For a more concrete example, a more standard cone can be constructed as $$ \text{Cone} = (S^1 \times [0,1]) / ((x,t)\sim (x',t') \iff t = t' = 1), $$ where $S^1$ is the unit circle in $\mathbb{R}^2$. Imagine a one-legged pair(?) of pants with a drawstring waist. You start with a cylinder of fabric ($S^1 \times [0,1]$), and then pull the string tight so that the waist closes down to a point. This is the cone.