# Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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### Test if line segment formed by two points intersect with image edges

I have a map that looks roughly like the following picture. The map is discretized into pixels and has size nxn where n is in <...
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### Nagata-Smirnov vs. Urysohn metrization theorems - an example?

I'm looking for an example to demonstrate that fact that the Nagata-Smirnov thm is "more useful" than Urysohn's. That is, I'm looking for a space that you can prove is metrizable using Nagata-Smirnov ...
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### Given subset $A$ and $A_x = \{y : (x,y) \in A\}$, Are the following equivalent

(This is Baire-Fubini theorem for categories) For a subset $A \subset \mathbb R^2$ and $x \in \mathbb R$ we denote $$A_x = \{y : (x,y) \in A\}$$ Are the following equivalent? $A$ is of first ...
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### Is a compact metric space with a “doubled” point sequentially compact?

Let $X$ be a compact metric space with topology $\tau$ generated by the metric. Consider a new point $x_1\notin X$ and a non-isolated point $x_0\in X$. Set $\overline{X}=X\cup \{x_1\}$, equipped with ...
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### The Fundamental group of the circle from “Introduction to knot theory”, Ralph H. Fox (3)

In this book,as a third completion (the second completion I have not asked it yet as I am still writing it) to The Fundamental group of the circle from "Introduction to knot theory", Ralph H....
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### Proving that $(0,1)^3$ not homeomorphic to $[0,3)^3$

What are some of the various ways of proving that $(0,1)^3$ is not homeomorphic to $[0,3)^3$ using the fundamental group and homology groups? I feel like I have various ways of understanding why this ...
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### Is any space with this property homeomorphic to the three-torus? [on hold]

If you have a three-dimensional, locally Euclidean space such that any path along a coordinate direction is closed (you always eventually come back to your starting point), is this necessarily ...
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### How do we prove that this set is not manifold? [on hold]

Prove that: $$Z:=\Big[ {(x,y,z) in R^3|x^2 +y^2-z^2=0}\Big]$$ is not a manifold.(Even It is not a topology manifold )
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### Every open set is a countable union of compact sets. [In which kind of topological spaces this is true]

Let $X$ be a Hausdorff topological space. Under which hypotheses on $X$ every open subset can be written as a union of countably many compact sets. I was wondering if $\sigma$-locally compact is a ...
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### The Fundamental group of the circle from “Introduction to knot theory”, Ralph H. Fox (1)

The says in the beginning of discussing this title: "Let the field of real numbers be denoted by R and the subring of integers by $J$. We denote the additive subgroup consisting of all integers ...
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### Homotopy equivalence of S1 and R2-0 [duplicate]

I want to show that $X=S^1=\{x^2+y^2=1|x,y\in \mathbb{R}\}$ and $Y=\mathbb{R}^2-\{0\}$ are homotopy equivalent. For this I have to find a function $f:X\rightarrow Y$ and a function $g:Y\rightarrow X$ ...
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### Which subsets of $\Bbb R$ are a countable union of open sets and countable sets?

From (1), every open subset of $\Bbb R$ is at most a countable union of open intervals. The converse is also true: any countable union of open intervals is an open set. However, I want every subset of ...
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### Co-Countable topology cases

Let X be the reals. The $\tau$ $=$ $\{$ $\varnothing$ $\}$ $\cup$ $\{$ $A$ $:$ $A^c$ is countable $\}$ What I want to show is that the union of an arbitrary collection of open sets is open. So I ...
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### Elementary question on Hausdorff distance: Difference between integrals on level sets

Let $U \subset \mathbb R^2$ be compact and consider $h:U \times (0,T) \to \mathbb R$ a uniformly bounded function such that: $0 \lt c_1 \le h \le 1-c_2$ $f_1,f_2 \in W^{2,1}_p(U \times (0,T))$ for ...
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### countable family of open sets

Is there a countable family of open subsets of ${\bf R}$ or $[0,1]$ such that each rational belongs to only finitely many of the open sets and each irrational belongs to infinitely many of the sets?