1
vote
1answer
40 views

“Broken-line paths” in $\mathbb R^n - \{ 0 \}$

In Munkres's Topology, he says: Suppose $x$ and $y$ are two different points from zero of the punctured euclidean space $\mathbb{R}^n -\{0\}$. We can join them a path by the straight-line path ...
0
votes
1answer
41 views

What is a “closed subspace” of a topological space?

I was reading a proof online and it linked to a book by Munkres which says Every closed subspace of a compact space is compact. I dug out the book and searched the index for this term. ...
0
votes
1answer
59 views

Is there a name for continuous functions $\Omega \rightarrow \mathbb{R}$ that can be continuously extended to $\overline{\Omega}$?

Given topological spaces $X$ and $Y$ together with a subset $\Omega \subseteq X$, is there a name for those continuous functions $f : \Omega \rightarrow Y$ such that $f$ can be extended to a ...
0
votes
0answers
30 views

Can we call the boundary of a subset of a topological space “partial X”?

Intuitively, one might be tempted to say $\partial S$ (the boundary of $S\subseteq X$ for X a topological space) as "partial X". Is this formally valid?
0
votes
1answer
58 views

Is there a name for this variant on “continuous function”?

Let $X$ and $Y$ denote topological spaces. Then a function $f : X \rightarrow Y$ is said to be continuous iff for all $U \in \mathcal{P}(Y)$, it holds that if $U$ is open in $Y$, then $f^{-1}(U)$ is ...
0
votes
0answers
16 views

How to call a function defined on a set with gaps on arbitrarily small scales.

Let $I$ be an interval and $A\subset I$ such that for any two points $x,x'\in A$ there exists an interval $J$ between $x$ and $x'$ such that $J\cap A=\emptyset$. How does one call this proerty of ...
0
votes
0answers
33 views

What does “single set” mean in this context?

I encountered this problem in Munkres topology. Let $X_1 , X_2$ denote a single set in topologies $\tau_1$ and $\tau_2$, respectively; let $Y_1 , Y_2$ denote a single set in the topologies $U_1, ...
1
vote
2answers
66 views

How to call two subsets that can be deformed into each other?

Given a topological space $X$, is there a canonical name for the equivalence relation generated by the following relation on the subsets of $X$? $A \sim B :\Leftrightarrow \exists \text{ continuous } ...
6
votes
1answer
55 views

Closed and Open Set - History of Terms

I know very little in the way of math history, but I question that was bothering me recently is where the terms open and closed came from in topology. I know that it's easy to ascribe a sense of ...
3
votes
1answer
49 views

Sets that are equal to closure of its interior

Is there a standard name for set $M$ for which $M = \overline{M^0}$? $M^0$ is interior and $\overline{M}$ is closure. Often I work with well behaved sets and functions. I have in mind continuous ...
0
votes
1answer
51 views

Has this topology a name?

let $(X,\tau )$ be the topological space where $\tau =\{\emptyset, X, \{x\}, X-\{x\}\}$ , $ x \in X$. Does this topology have a name? Thanks in advance!
3
votes
1answer
45 views

Local topological properties

Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of ...
0
votes
4answers
60 views

Why to see that $\overline{B}(x;r)$ is closed if it was just defined?

I'm reading Conway's A Course in Point Set Topology. He defines open and closed balls and then he introduces some examples, one of these examples is this: (c) For any $r>0$, $\overline{B}(x;r)$ ...
1
vote
1answer
144 views

two interlocked circles are homeomorphic to two noninterlocked circles

This is what I learned from here the post: two interlocked circles are homeomorphic to two noninterlocked circles, thus they (two interlocked circles and two noninterlocked circles) are homotopic ...
1
vote
3answers
81 views

What is a co-dimension?

I'm looking for a simple explanation (without complex formula) what a co-dimension is. When does objects have a co-dimension of 0 and when > 0? Context: A Critical Comparison of the 4-Intersection ...
5
votes
1answer
129 views

What exactly is a dimension?

Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a ...
0
votes
1answer
61 views

Is there a name for the one-point compactification of $\mathbb{C}$?

Let $\hat{\mathbb{C}}$ be the one-point compactification of $\mathbb{C}$. This space $\hat{\mathbb{C}}$ is called the Riemann sphere. If I want to designate the topology $\tau$ on ...
1
vote
1answer
67 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
0
votes
4answers
94 views

How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
1
vote
1answer
56 views

Is there a name for a topological space $X$ in which Every closed subset $A\subsetneq X$ is contained in a countable union of compact sets

As was recommended for me in here I would like to share the following question with you: Is there a name for a topological space $X$ which satisfies the following condition: Every closed subset ...
2
votes
1answer
56 views

Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is contained in a countable union of compact sets Does Baire space satisfy this ...
0
votes
1answer
35 views

Is there any standard terminology for the quotient of a topological group by the connected component of the identity?

If $G$ is any topological group, then the connected component of its identity is a closed normal subgroup $H$. It follows that $G/H$ is a totally disconnected topological group. Often, $G$ will be ...
0
votes
0answers
35 views

terminology: accumulation points, limit point, cluster point

In a topological space $X$, what would be the most common terms to describe the following two properties about a point $x\in X$ and a subset $S\subseteq X$. I) For every open set $U$ with $x\in U$, ...
1
vote
0answers
15 views

Terminology for homotopies which stay inside some finite stage of a union

Sometimes it happens that you have a sequence of topological spaces each contained in the next $$ X_1 \subset X_2 \subset X_3 \subset \ldots$$ and you want to talk about things like homotopy in the ...
2
votes
3answers
188 views

Path components or connected components?

Can anyone explain the difference between these two terms? Are they basically different names for the same thing or totally different things?
2
votes
0answers
41 views

What are the names for the structures obtained when we drop some topological space axioms?

Motivation: If I start with the group axioms and drop the requirement that I have inverses, I get the monoid axioms. If I proceed to drop the requirement that I have an identity, I get the semigroup ...
7
votes
3answers
571 views

Why are compact sets called “compact” in topology?

Given a topological space $X$ and a subset of it $S$, $S$ is compact iff for every open cover of $S$, there is a finite subcover of $S$. Just curiosity: I've done some search in Internet why compact ...
3
votes
0answers
95 views

“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 ...
2
votes
1answer
43 views

terminology needed

This is just a terminology question. Let $Y$ be a topological space. Is there a word to describe those topological spaces $X$ that contain $Y$ as a dense subspace? If not, what would you call such ...
0
votes
0answers
49 views

Quasicompact? Why the distinction?

What is the reason that some topologists use quasicompact? Why is the distinction made? quasi means "not really", so why use this terminology?
2
votes
0answers
44 views

What is the name for the topology where every point is in the boundary of an open set?

Is there a name for topological spaces in which every point is in the boundary of an open set?
0
votes
1answer
91 views

How to make a ghost manifold [closed]

How does one mathematically define a manifold that can pass through another manifold? A "ghost" passing through a "wall" type construction. I understand that this may be done by creating a copy of the ...
2
votes
0answers
44 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
1
vote
1answer
46 views

The “standard” definition of discrete subspace

Let $X$ be a topological space and $D\subset X$ a generic subspace. In literature I have found the following different definitions: $D$ is discrete in $X$ if $X$ doesn't contain limit points of $D$. ...
1
vote
1answer
53 views

If $B$ is a cover of $A$, then do we say that $C$ is a subcover of $B$, or of $A$?

My textbook variously says both "subcover of $A$" and "subcover of $B$" to refer to a subcollection $C$ of the collection $B$ (that covers $A$); is this usage standard and is it not potentially ...
7
votes
0answers
121 views

Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?

A topological space $\langle X,\tau\rangle$ is said to be normal if any two disjoint closed subsets are separated by open sets, meaning that for disjoint $E,F\subseteq X$ with $X\setminus E,X\setminus ...
8
votes
5answers
330 views

Motivation for the concept of “open set” in topology

I am looking at the section "Motivation" for the Wikipedia entry on "open sets": https://en.wikipedia.org/wiki/Open_set#Motivation and I am not sure it is doing such a good object of motivating open ...
0
votes
5answers
338 views

Why do we say a real line is an open set, but the complex plane is not an open set?

Why do we say a real line is an open set, but the complex plane is not an open set? I don't understand that? I'm fully confused between these differences?
0
votes
3answers
97 views

Is there a word for the classification of a set as continuous or discrete?

For example, in computer science, there can be zero, one, two, etc. parameters to a computer program, and this is called its "arity". Sets can be countable or uncountable. Is there some word I can use ...
3
votes
0answers
68 views

Is there a name for the following property in the theory of topological spaces?

I worked out the following lines. Probably it is totally trivial, but maybe this is interesting in the theory of topological/metric spaces. Let $\Omega$ be an open subset of the Euclidean $n$-space. ...
1
vote
1answer
53 views

What do we call a continuous function that induces a homeomorphism onto its image?

I know that an order-homomorphism that induces an isomorphism onto its image is called an order-embedding. So, I was expecting that a continuous function that induces an homeomorphism onto its image ...
2
votes
2answers
74 views

Please help me check my metric definition of isolated point

I translated the word definitions into the more symbolic form below, but as they aren't mere negations of each other, it was a little tricky. Is there any mistake below (especially for 'isolated ...
0
votes
2answers
608 views

Definition of accumulation point

I have here a definition of accumulation point: A point $x$ in a metric space $M$ is called an accumulation point of $A \subset M$ if every neighbourhood of $x$ contains some point of $A$ distinct ...
0
votes
2answers
99 views

How to define an interior point in terms of $\epsilon$-balls?

Which is the technically correct definition? I) An interior point of a set $B$ is a point that is the centre of some $\epsilon$-ball in $B$. II) An interior point of a set $B$ is a point that is in ...
2
votes
2answers
423 views

Precise definition of epsilon-ball

My textbook gives the following definition: "For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$." Is this correct? ...
3
votes
1answer
75 views

Does a neighbourhood need to be a *connected* set?

I have in my topology/ real analysis textbook the definition of neighbourhood of a point as an open set containing that point. But isn't a neighbourhood necessarily a connected set? Wikipedia also ...
1
vote
1answer
100 views

What is a finite partial function from $\mathfrak c$ to $D=\{0,1\}$

While I reading a paper, there is a notation is called finite partial function. I searched by google, I cannot find its definition. So I post it here as a question: What is a finite partial ...
2
votes
2answers
118 views

What does it mean to “identify” points of a topological space?

I was recently reading about circle rotations (a basic example in dynamical systems) and got confused by some notation. It said consider the unit circle $S^{1} = [0,1]/{\sim}$, where $\sim$ indicates ...
12
votes
1answer
160 views

a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
0
votes
1answer
32 views

How could I define this $\mathrm{nw}(X)$ by using only one sentence?

A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any neighbourhood $U$ of $x$ there exists an $M \in \mathcal N$ such that $x\in M ...