# Tag Info

## Hot answers tagged general-topology

Accepted

### Are rational points dense on every circle in the coordinate plane?

They're not. No two different circles centered at the origin contain any of the same points. There are uncountably many circles (specifically, one for each real number, corresponding to the radius) , ...
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### How can a mug and a torus be equivalent if the mug is chiral?

Other answers have said that topology cannot distinguish chirality, but this is not quite true. Topologically a left and right glove are identical, because the left-right reflection is continuous in ...

### Examples of the difference between Topological Spaces and Condensed Sets

Here's a one-paragraph answer: Topological spaces formalize the idea of spaces with a notion of "nearness" of points. However, they fail to handle the idea of "points that are ...
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### "Naturally occurring" non-Hausdorff spaces?

The digital line is a non-Hausdorff space important in graphics. The underlying set of points is just $\mathbb{Z}$. We give this the digital topology by specifying a basis for the topology. If $n$ ...
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### Explain this mathematical meme (Geometers bird interrupting Topologists bird)

The topologist bird is stating some nice simple topological concepts. The geometer bird comes along and interrupts the topologist bird, squawking loudly over the top with something much longer, more ...
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### Can path connectedness be defined without using the unit interval?

There are really two separate questions here: can you define the unit interval space without talking about real numbers, and can you define path-connectedness without talking about the unit interval ...
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### Is differentiation as a map discontinuous?

For a counterexample, take the sequence $$\frac {\sin nx} n$$ These are all continuously differentiable, but the sequence converges to $0$ and the sequence of derivatives doesn't converge at all. The ...
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### Is there a formulation of topology which excludes most of the pathological objects?

Reading about the history of topology was enormously clarifying for me here; I can warmly recommend checking out James' History of Topology and/or Dieudonné's History of Algebraic and Differential ...
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### Closure of the union = Union of closures

(1) ($\supset$) :: \begin{align*} A \subset A \cup B \implies \text{cl}(A) \subset \text{cl}(A \cup B) \\B \subset A \cup B \implies \text{cl}(B) \subset \text{cl}(A \cup B) \end{align*} therefore ...
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### Difference between topology and sigma-algebra axioms.

I would like to mention that in An Epsilon of Room, remark 1.1.3, Tao states: The notion of a measurable space (X, S) (and of a measurable function) is superficially similar to that of a ...
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### Open maps which are not continuous

There is in fact a rather easy example of a function $\mathbb R \to \mathbb R$ such that the image of every open set is $\mathbb R$: Let $(x_i)_{i\in\mathbb Z_+}$ be the binary expansion of $x$, so ...

### Possible number of open sets in a topology

Why not the topology $$\mathcal T=\big\{\varnothing ,\{1\},\{1,2\},\{1,2,3\},\{1,2,3,4\},...,\{1,2,...,99\}\big\},$$ on the set $E=\{1,...,99\}$ ?
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### Is every open set the interior of a closed set?

The set $(0,1) \cup (1,2)$ is a counterexample.
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### Why is a discrete topology called a discrete topology?

The discrete topology has a topological structure which perfectly reveals the discrete nature of the underlying set of points You can consider a set to be a discrete collection of objects. To a given ...
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### Why don't we use closed covers to define compactness of metric space?

It is important to understand that, although definitions often look arbitrary, they never are. Mathematical objects are intended to model something, and you can't understand why the definition is the ...