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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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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140 views

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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103 views

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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60 views

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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Example of subsets in plane with continuous bijective mapping between them

The question is from C. Pugh's Real Analysis: Construct nonhomeomorphic connected, closed subsets A, B $\subset$ $\mathbb{R}^2$ for which there exists continuous bijections $\;f: A \to B$ and $\;g: B\...
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201 views

How to quantify the “uniformity” of a distribution of holes in a surface

I want to try to quantify if the distribution of holes over a surface is uniform or not. The holes can have any given shape and can be arranged in any way over the surface. Three examples are ...
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128 views

Topology on $\text{Homeo}(X)$ Which Captures Topological Group Actions.

Definition. Let $G$ be a group and $X$ be any set. We may define a group action of $G$ on $X$ as map $\cdot: G\times X\to X$ such that $e\cdot x=x$ for all $x\in X$ and $g\cdot(h\cdot x)=gh\cdot x$ ...
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Neighborhood Topology and Open Set Topology: their Equivalence and Comparison

Motivation. A few years ago we were using Armstrong's Basic Topology as a textbook for the topology course in my university, and right off the bat I had a huge conceptual problem regarding the two ...
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137 views

Connected topological manifolds

For any connected topological manifold, it is true that for any two points on the manifold, there exists a single local chart that both of two points lie in it. How can I prove it?
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Gluing Together Borel Measures

Does anyone know a standard reference for the following, which I assume is true: X a topological space, $\{U_i\}$ an open cover, $\mu_i$ a collection of regular Borel measures agreeing on overlaps. ...
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Space of cadlag functions - Nonexistence of a TVS Polish topology?

Consider the space $D := D([0,1], \mathbb{R})$ of real-valued cadlag functions on $[0,1]$. On $D$ one can invent several (more or less) well-known topologies: the uniform topology $U$: $(D, U)$ is a ...
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Limiting the size of near-coherence classes in $\omega^*$

We say that two ultrafilters in $\omega^*$ (i.e., two non-principal ultrafilters on $\omega$) are nearly coherent if there is a finite-to-one mapping $\varphi:\omega\to\omega$ such that $\beta\varphi$ ...
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547 views

Proof of the Arzela-Ascoli theorem - where is the assumption that $X$ is compactly generated used?

I'm learning the proof of the following version of Arzela-Ascoli's theorem (Willard, General Topology, page 287): Let $X$ be a Hausdorff, or regular, k-space, $(Y,\mathcal D)$ a Hausdorff uniform ...
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Prob. 1, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?

Here is Prob. 1, Sec. 27, in the book Topology by James R. Munkres, 2nd edition: Prove that if $X$ is an ordered set in which every closed interval is compact, then $X$ has the least upper bound ...
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Regarding the connected component of $|1/J| < 1$ containing $\infty$

How does one explicitly describe the connected component of $|1/J| < 1$ containing $\infty$? Here, $J = J(\tau) = j(\tau)/12^3$ is the normalized $j$-invariant so that $J(i) = 1$, and $\tau$ is in ...
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265 views

projective space and torus

we defined the projective space as $\mathbb{S^2/Z_2}$ i.e. identify antipodal points and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
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179 views

Order and Metric

Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ...
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Prob. 8, Sec. 19 in Munkres' TOPOLOGY, 2nd ed: What is the situation in the box topology?

Let $\mathbb{R}^\omega$ denote the set of all sequences of real numbers, and let $(a_1, a_2, a_3, \ldots ), (b_1, b_2, b_3, \ldots) \in \mathbb{R}^\omega$ be fixed with $a_i > 0$ for all $i= 1, 2, ...
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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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216 views

Hermitian Matrices with At Most Pair-wise Eigenvalue Degeneracy

Let $n\in2\mathbb{N}$ be given. Let $H\in Mat_{n\times n}(\mathbb{C})$ be a Hermitian traceless matrix such that its eigenvalues have at most pairwise degeneracy. (That is, if the eigenvalues are $\{\...
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365 views

When does pointwise convergence on compact space imply uniform convergence?

I just wondered whether there is a more general theorem behind claims like 'if a sequence of equicontinuos functions $f_i:[a,b]\rightarrow{\bf R}$ converges pointwise to a continuous function $f$ then ...
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What is the topological interpretation of continuity of distributions?

I was given this definition of continuity in the distributional sense. A distribution $T$ over the space of test functions $\mathcal{D}$ is continuous if for every sequence of test functions $\{ \...
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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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191 views

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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(Topological Groups) Show that $g_{\alpha}(x) = x *\alpha$ is a homeomorphism of G.

Note: This is not homework. Can someone please verify my proof or offer suggestions for improvement? Let $\alpha$ be an element of a topological group $G$. Show that the map $g_{\alpha}: G \...
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Equivalence of Lebesgue Measurablity

Hello Mathematics Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am ...
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Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
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163 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
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In which of the three topologies is $f(t)=(t, 2t, 3t, 4t, \ldots)$ continuous? Here, $f$ is a function from $\mathbb{R}$ to $\mathbb{R}^\omega$.

Can someone please verify my proof or offer suggestions for improvement? Consider the product, uniform, and box topologies on $\mathbb{R}^\omega$. In which of the three topologies is $f(t)=(t, 2t, ...
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Show that $\overline{A} \cup \overline{B} = \overline{A \cup B}$.

Can someone please verify my proof? Show that $\overline{A} \cup \overline{B} = \overline{A \cup B}$. Clearly, $$A \cup B \subseteq \overline{A \cup B}$$ So, $$A \subseteq \overline{A \cup B}$$ $$...
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Properties of first-countable spaces

Hi I have questions regarding first-countable spaces. I just want to confirm something: The following are properties regarding limits and continuity of first countable spaces on Wikipedia: If $f$ is ...
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410 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. f\...
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Infinite Dimensional Topological Vector Space

Let $V$ is a finite-dimensional vector space over $\mathbb{R}$ (or $\mathbb{C}$). To make $V$ a topological space, we may choose the sets $f^{-1}(U)$ as a sub-basis, where $f$ ranges over all linear ...
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Does $\pi_0$ commute with sequential colimits?

I'm looking for an example of a sequence of topological spaces $Y_1 \rightarrow Y_2 \rightarrow \cdots$ such that the induced map $$ \text{colim}\, \pi_0(Y_i) \rightarrow \pi_0 (\text{colim}\, Y_i) $$ ...
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How to show that the Tychonoff product is associative?

Let $\{ X_t : t \in T\}$, be a family of topological spaces. Suppose thst $T = \bigcup \{ T_s : s \in S \}$, where $T_s \neq \emptyset $ for all $s \in S$, and $T_s \cap T_{s'} = \emptyset$ if $s \...
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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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509 views

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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103 views

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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398 views

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 ...
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377 views

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 ...
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Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?

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 ...