Tagged Questions

1
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1answer
26 views

$\tau_1$ coarser topology than $\tau_2$

I know the following definition of coarser topologies: If $\tau_1$ and $\tau_2$ are two topologies on $X$, we say $\tau_1$ is coarser than $\tau_2$ if $\tau_1\subseteq\tau_2$. In my book about ...
0
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1answer
33 views

Proposition from the book “Convex analysis and measurable multifunction ”

Please , what is $F_{\sigma}$? And all $U$ can be writen as $\cup F_n$, or only open set ? Thank you
5
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0answers
87 views

I think a definition is wrong in “Model Categories” by Hovey.

I am working through the book "Model Categories", by Mark Hovey, and have a doubt about a definition given there. At the beginning of page 50 we read: Define a map $f:X\rightarrow Y$ to be a ...
2
votes
2answers
64 views

Is an “open system” just a topological space?

I think of an "open set" as being "roomy" or "spacious," in the sense that around every point, there is a little bit of room. This motivates the following definition. Definition. An "open system" ...
3
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0answers
45 views

Understanding the roots of homomorphism and homeomorphism

I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
3
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2answers
63 views

two notation: semi-metric and pesudometric

There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different? Thanks ahead.
1
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1answer
50 views

Limit point definition

I have read the definition of a limit point of a set in Real Analysis. The definition goes like: A number $p$ is said to be a limit point of a set of reals, $S$, if every neigbourhood of $p$ has at ...
6
votes
2answers
78 views

Collecting definitions of continuity.

Let $X$ and $Y$ denote topological spaces and consider a function $f : X \rightarrow Y$. I'm collecting possible definitions/characterizations of the statement "$f$ is continuous." Here's two to get ...
8
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3answers
210 views

What's Geometry?

I am a grad student. I am writing an article on geometry and relativity theory and trying to start with discussing basic ideas of topology. In my article I tried very hard to motivate the idea of ...
1
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3answers
76 views

Difference between closure and the boundary

I'm having a hard time distinguising the difference between the boundary and the closure of sets. They seem so similar, but that almost sounds too good to be true. So if the boundary is just the ...
1
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0answers
32 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
4
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1answer
53 views

questions on the completely accumulation

Could somebody help me to understand the definition of completely accumulation? And help me show that this claim: A space $X$ is compact iff every infinite set in $X$ has a point of complete ...
1
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1answer
71 views

Metrics with infinite distances.

I've been wondering about the spaces $\Bbb R\cup\{-\infty,+\infty\}$ and $\Bbb C\cup\{\infty\}.$ Is there a useful generalization of the definition of a metric they satisfy? I thought it would be ...
2
votes
2answers
41 views

Understanding the definition of monotonically monolithic

A collection $\mathcal{N}$ of subsets of $X$ is called an external network of $A \subset X$, when for every $x \in A$ and every neighbourhood $U$ of $x$, there exists some $N \in \mathcal{N}$ such ...
8
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1answer
258 views

Semi-Norms and the Definition of the Weak Topology

When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ...
2
votes
2answers
78 views

What is the relation between $ \kappa$-monolithic and monotonically monolithic?

For an infinite cardinal $\kappa$, a space $X$ is called $\kappa$-monolithic if $nw(\overline{A}) \le \kappa $ for any set $A \subset X$ with $|A| \le \kappa$. And you can see this definition of ...
1
vote
2answers
207 views

What are relative open sets?

I came across the following: Definition 15. Let $X$ be a subset of $\mathbb{R}$. A subset $O \subset X$ is said to be open in $X$ (or relatively open in $X$) if for each $x \in O$, there exists ...
5
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3answers
79 views

Understanding a definition of radial

A space $X$ is called radial if, for any $A \subset X$ and any $x \in cl(A)$, there is a transfinite sequence $s=\{a_\alpha: \alpha \in \kappa\} \subset A$ which converges to $x$. What's meaning of ...
2
votes
1answer
81 views

What does compact cover mean?

I am reading a difinition of Lindelof $\Sigma$ space. It talked about compact cover. As the title explains, what does compact cover mean? It means every member of the cover is compact?
3
votes
1answer
76 views

A question on linear ordered space

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$. And we know every space contains a dense left-separated subspace. My question ...
1
vote
2answers
123 views

Definition of a basis for a particular topological space

I'm currently looking at Lemma 13.2 in Munkres' Topology. It states the following: Given a collection $C$ of open sets of a topological space $X$ such that for each open set $U$ of $X$ and each $x$ in ...
5
votes
4answers
278 views

Can open sets be an open cover, for itself?

I have Baby Rudin's book with me and it clearly defines a cover to be open. In a followup, it defines a set $K$ to be compact if every open cover of $K$ contains a finite subcover. And the rest I ...
2
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1answer
134 views

Base (topology) with closed intervals

I am curious why it's a problem to define a base using closed sets? For example, my book uses the definition under "Constructing Topologies from Bases" as specified at ...
1
vote
1answer
61 views

What's the meaning of $C$-embedded?

What's the meaning of $C$-embedded? It is a topological notion. Thanks ahead.
1
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1answer
96 views

What is the meaning of the term “inductively P map”?

In this page is the definition of an inductively open map. But in this pdf is the definition of a inductively P map, where P is a property of maps. But there is a difference in the definitions. In ...
3
votes
3answers
212 views

Spectrum of a field

Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals. Thanks a ...
2
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1answer
89 views

Is there a name for a collection of open sets where arbitrary intersections are open?

Let $\mathcal{U} = \{U_i\}_{i\in I} $ be a collection of open sets with the property that the set $\bigcap_{i\in J} U_i $ is open for all subsets $J$ of $I$. Is there a name for such collections of ...
17
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3answers
475 views

Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals

I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have ...
2
votes
1answer
105 views

The precise definition of a “sheaf of rings”

Let $X$ be a topological space. Let $Op(X)$ be the category of open sets. A sheaf of abelian groups is a (contravariant) functor $F:\mathcal{C} \to Ab$ satisfying the sheaf condition: the following ...
1
vote
2answers
109 views

Elaborate on $A^{c}:=\{p\in\mathbb Q : 0<p<\sqrt{2}\}$ not open and not closed in $\mathbb R$

I know that $\sqrt{2}\not\in\mathbb Q$ and $\sqrt{2}\in\mathbb R$ but it is not obvious to me why $\{p\in\mathbb Q : 0<p<\sqrt{2}\} \subset \mathbb R$ is not open. If it is not open, it means ...
1
vote
3answers
58 views

A parametrized surface

If I am given a map $f:U\subset \mathbb R^2 \to \mathbb R^3$ where $(x,y)\to (f(x,y),g(x,y),h(x,y))$. Is this necessarily a "parametrized surface"?  Am I right in thinking that any map of the above ...
0
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1answer
110 views

Definition of Genus

Are genus necessarily toral -- as shown in the illustration on wiki what about a tube, does it qualify for having genus 1? What about this? Does this have genus 1 or 2? Thanks.
1
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1answer
172 views

Does perfectly normal $\implies$ normal?

A question here on perfectly normal spaces got me into investigating the definition of such a space. The definition on wikipedia says A perfectly normal space is a topological space X in which ...
3
votes
4answers
376 views

what exactly is an open set?

Many, infact all the books on topology I have come across define open sets in the following way: "A set $A$ is said to be open if by moving in small amounts in any direction about any point we ...
5
votes
1answer
129 views

What motivates discrepancies between the definitions of “continuous” and “limit”?

I am working from Munkres' Analysis, and I've converted his definitions slightly to make them easier to compare. In the table below, you can fill in the blanks in the top row with words from the ...
14
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5answers
1k views

Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this? I'm assuming ...