# 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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### Equivalence (Or non-equivalence) of definitions for Metric Spaces

So, I've stumbled across a definition of a metric space that has given me some pause. The definition of a Metric Space that I've always used has been the following: A Metric Space is a set $X$ ...
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### Image of compacts sets under a pseudo-continuous functions is bounded

I'm reading a paper where the author makes the following remark: If $K\subseteq 2^\omega$ is compact and $\phi:K\rightarrow\omega^\omega$ is pseudo-continuous, then $\phi''K$ is bounded. Let me put ...
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### $h_*:S_*(X\times Y)\to S_*(X\times Y)$ a homomorphism of natural chain complexes. If $h_0=Id$, show that $h_*$ is homotopic to $Id$

Let $X$ and $Y$ be topological spaces and $h_*:S_*(X\times Y)\to S_*(X\times Y)$ a homomorphism of natural chain complexes. If $h_0=Id$, show that $h_*$ is homotopic to $Id$. To solve this problem I ...
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### Is any open set of $(0,\infty)\times S^{k-1}$ a countable disjoint union of products $I\times A$, where $I$ is an interval and $A$ is open?

Let $V$ be an open set in $(0,\infty)\times S^{k-1}$. I want to prove that $V$ can be written as a countable disjoint union of the form $$V=\coprod_{n=1}^\infty I_n\times A_n,$$ where each $I_n$ is an ...
1answer
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### Application of hyperspheres

Recently I've been studying the the volume of an n-ball. Do hyperspheres (or their volume/surface formulas) have any real-world applications?
1answer
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### Topology of derived sets and subset [on hold]

if A is a subset of C ,then A' is a subset of C' A' denotes set of all limits points of A
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### A product of two connected spaces is connected.

I know the standard proof of this theorem, but is there any problem with this proof: Let $\varnothing \neq E \subseteq X \times Y$ then $E=\bigcup U_\alpha \times V_\alpha$ where $U_\alpha, V_\alpha$ ...
1answer
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### Space of Bounded Functions

Let $B(\mathbb{R},\mathbb{R})$ be the space of all bounded functions on $\mathbb{R}$. Is it possible to define two norms on this space generating two different topologies? The only norm I know on ...
1answer
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### Index of a point on a Jordan curve

Let $p$ be a point on a Jordan curve $J$. Let $\gamma$ be a loop in the bounded component of $\mathbb{C} \backslash J$. Show that $ind(\gamma,p) = 0$. Can we just tell that the curve is contractible ...
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### Is it possible to make a stereographic projection of a stereographic projection?

Since we can't visualize a 3-sphere directily, I was wondering how does its stereographic projection look like, and found this neat demonstration: https://demonstrations.wolfram.com/...
1answer
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### Regarding Urysohn's metrization theorem proof

At Munkres' topology, he showed two different proof of Urysohn's metrization theorem. In the first version of the proof, he tried to construct some continuous function $F$ from a given space $X$ ...
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### A question about non-wandering points and their definition

As I read in a paper the definition of non-wandering points is this : $x \in X$ is a non-wandering point if for every neighborhood $U$ of $x$ there exists a natural number $n$ suchthat : \...
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### If $p$ is a regular covering then so is $q$.

If $q:X\to Y$, $r:Y \to Z$, and $p=r \circ q : X \to Z$ are all covering maps, with $Z$ locally path-connected, and if $p$ is a regular covering then so is $q$. Note 1. The condition $q$ is a ...
2answers
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### Continuous Bijection of Top Spaces

It is well known that a continuous bijection of compact hausdorff topological spaces is a homoemorphism. I am wondering, is it true that a continuous bijection of compactly generated spaces is a ...
2answers
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### How big can the Hausdorff dimension of the closure of a smooth curve be?

Consider curves in $\mathbb{R}^n$. Smooth curves have Hausdorff dimension $1$. The closure of a smooth curve can have Hausdorff dimension $> 1$. (For example, a curve dense in a torus.) How big ...
1answer
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### Definition of finite topology? [on hold]

Consider topology $T$ on $X$ where $X$ is a set given to us. When do we say topology is finite? When $X$ is finite? or when $T$ contains finite number of open sets? or both when $X$ is finite as ...
2answers
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### Is the following set topology?

Consider a potential topology $T$ on $\mathbb R^3$ (three dimension space.) $T$ contains sets $$\{(x,y,z) \mid x^2 + y^2 + z^2 \leq r\},$$ where $r\in\mathbb R^{\geq 0}.$ Since we know that that ...
1answer
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### A Direct Proof that a Topological Space with Ininitely Many Components is Disconnected

It's straightforward to prove that for a topological space, connected $\iff$ having one component. It then follows by contra-positive that disconnected $\iff$ having more than one component. In ...
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### Is Sorgenfrey Line a discrete space?

The base of Sorgenfrey line consists of all sets of the form [a,b). By this all sets of the form [a,b) are clopen. How can we prove that the Sorgenfrey line is not discrete i.e. that the singelton {x} ...
1answer
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### To prove topologically not same

The sets $A=\{ (x,y) \in \mathbb{R} \;|\; xy=0 \text{ and } x+y \geq 0 \}$ $B=\{ (x,y) \in \mathbb{R} \;|\; xy=0 \}$ are not homeomorphic. In $A$ if we remove the origin it becomes $2$ pieces, ...
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### In general topological spaces (X,T) is it true that “S is with no isolated point = Every point of S is a limit point of S”?

In general topological spaces (X,T) (not only in metric spaces), let S be a non-empty subset of X, is it true that "S is with no isolated point = Every point of S is a limit point of S"? It seems ...
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### Definitions for regular submanifold in terms of level sets?

From this question, we have: Let $S$ be a subset of a smooth $n$-manifold $M$. Then $S$ is an embedded $k$-submanifold of $M$ if and only if Every point $p\in S$ has a neighborhood $U\subset M$ ...
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### Is the zero set of coordinate function a regular level set (regular zero set)?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
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### A basic question about projection map with quotient space

I have a true-false statement as follows: Suppose $X$ is a topological space with a quotient space $Y=X/{\sim}$ and projection map $p:X\to Y$. If $U\subseteq X$ is open, then $p(U)$ is open. I think ...
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### Verification of proof that if $S$ is connected then the closure of $S$ is connected.

I attempted to prove this theorem by contrapositive. Suppose $\overline{S}$ is disconnected. Then $\overline{S}=A\cup B$ such that $\overline{A}\cap B=A\cap \overline{B}=\varnothing$ and $A$ and $B$ ...
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### How do we define convergence of a sequence in $\mathbb R\cup\{-\infty,+\infty\}$

Can we define convergence of a sequence in the set $\mathbb R\cup\{-\infty,+\infty\}$?How do we define that?Do we require metric space for this convergence?
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### Show that a singleton $\{x\}$ is negligible

With the following definition of negligible set : $S \subseteq \mathbb{R}$ is negligible if \forall \varepsilon > 0 \hspace{0.2cm} \exists I_{k} : S \subseteq \bigcup\limits_{k \in \mathbb{N}}I_{...
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### Rational topological basis for Euclidean topology - topology without tears 2.2.3

Context: self-studying topology without tears, now at question 2.2.3. Question: Let $\mathcal{B}$ be the collection of all open intervals $(a,b) \in \mathbb{R}$ with $a \lt b$ and $a, b$ rational ...