# 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.

6,637 questions
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### Union of two locally compact space

Let $X=\mathbb{R^2}$ and let $$A=\{(0,0)\}\cup\{(x,y):x>0\}\subseteq X\;.$$ Notice that there is no compact set around $\{(0,0)\}$ because it will be open set. It looks like work for example of ...
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### Is the set of symmetric positive-definite matrices open in the set of symmetric matrices?

I'm not sure if the set of symmetric positive-definite matrices is open in the set of symmetric matrices. I'm almost certain that it is not open in the larger set of n by n matrices. So far most of ...
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### On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
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### Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator ...
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### Generalization of FTA

I'm sure that this is not any hypothesis, but following came to my mind when I was reading complex analysis. Consider a function $f(z)=z^n+g(z)$, where $g(z)$ is continuous (not necessary holomorphic)...
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### Zariski type topology in real algebraic geometry

Consider the following Zariski-type topology on $\mathbb{R}$: the closed sets are given by zero sets of real analytic functions. If we define a similar topology on $\mathbb{R}^n$, where the closed ...
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### Prove an annulus is homeomorphic to a cylinder

Let $A \subset \mathbb{R}^{2}$ be the annulus $A = \{(x,y) \in \mathbb{R}^{2} \colon 1 \leq x^{2} + y^{2} \leq 4 \}$. Prove that $A$ is homeomorphic to $S^{1} \times I$, where $I = [0,1]$ is the ...
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### Finding a good cover such that its lifting is still a good cover

Let $Y$ be a compact manifold, $X$ a topological space and $f: X \to Y$ a surjective map. Suppose further that every point in $Y$ has arbitrarily small open neighbourhoods such that their preimages in ...
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### Covering space of surface of infinite genus

Let $X$ be a surface of infinite genus that is not compact (with edges extending to infinity). How would I show that this is a covering space of the 2-torus $T^{1}\# T^{1}$ via the action of the free ...
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### Topological Spaces and Topological Equivalence

My professor gave us the following question while covering Topological Spaces Find a list of Topologies $\mathcal{T}_1, \mathcal{T}_2,..., \mathcal{T}_n$ on $X=\{1,2,3\}$ such that for every ...
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### Can $\operatorname{Spec}(A)$ be expressed as an inverse limit?

We know that given a ring $A$ such that $A/\mathfrak{R}$ is absolutely flat, then $\operatorname{Spec}(A)$ is Hausdorff (it's an equivalence). So $\operatorname{Spec}(A)$ becomes a quasi-compact, ...
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### Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
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### Definition of No Tear and No Paste

Topologists often mention an example beginning by "If there is no tear and no paste, then ...". As a student, I am confused with this "term", and I want to know the exact mean of it. First of all, ...
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### pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
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### Can open sets in different dimensions be homeomorphic to each other?

Assume $U$ is open in $\mathbb{R}^m$ and $V$ open in $\mathbb{R}^n$, $U\cong V$. Does it imply $m=n$?
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### Tangent bundles of smooth manifolds

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure ...
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### “clopen” terminology: acceptable?

I like the term "clopen" (a set which is both open and closed in a topological space), though an instructor of mine hated it when I used it recently. (Approximately, "never, ever use that again.") Is ...
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### Final-segment and finite-closed topology?

Suppose N is the set of all positive integers and t consists of N, the empty set, and every set { n, n+1, ... } for n any positive integer. This is a topology and is called the “final segment topology....
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### What kinds of structures support integration?

I am doing topology which neatly generalizes analysis, which led me to wonder naturally about generalizations of calculus. Specifically I'm interested in knowing what is required of a mathematical ...
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### Orientability as a topological property

Can one prove that orientability(of a manifold)is a topological property without using algebraic topology? That is, using a combination of general topology,linear algebra,and topological groups(such ...
### When should one use the two-point compactification of $\mathbb R$?
The real line $\mathbb R$ has a one-point compactification $\mathbb R\cup\{\infty\}$, where this "$\infty$" is at both ends of the line, so that the compactification is topologically a circle. It ...
The answer to Cardinality of a locally compact Hausdorff space without isolated points shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then \$|X| \ge ...