# Tagged Questions

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### What does it mean to say “Resolving intersections”

Consider a surface (with boundary) $S$ with marked points on the boundary such that we may may triangulate the surface. Call a line joining two marked points in a triangulation an arc. Consider a ...
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### 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 ...
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### 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 ...
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### 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!
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### 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 ...
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### 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 $r>0$, $\overline{B}(x;r)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $x\in U$, ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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?
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### 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?
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### 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 ...
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### 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: ...
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### 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 $D$. ...
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### 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 ...
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### Term for Sets Equivalent Up To Closure

Is there a nice name for the collection of sets that have the same closure? I'm writing up some notes developing some tricks for visualizing topologies, and I kind of want a non-cumbersome way to ...
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### What is the difference between isomorphism and homeomorphism?

I have some questions understanding isomorphism. Wikipedia said that isomorphism is bijective homeomorphism I kown that $F$ is a homeomorphism if $F$ and $F^{-1}$ are continuous. So my question ...
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### Terminologies related to “compact?”

A set can be either open or closed, and there can either be a finite or infinite number of them. A "compact" set is one where every open cover has finite subcover. Is there such a thing as a set ...
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### “Principal uniform space” vs “discrete uniform space”?

Which terms are better for a uniform space such that the set of entourages is a principal filter? "Principal uniform space" or "discrete uniform space"? "Principal uniformity" or "discrete ...
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### Replacing the value of a function with the value of the limit - is this a standard construction?

Consider a partial function $f : X \rightarrow Y$ where $X$ and $Y$ are topological spaces and $Y$ is Hausdorff. Note that, although the source of $f$ is $X$, the actual domain of $f$ is a (not ...
I'm looking for references (and a name!) for the following invariant of a topological space $X$: The least (infinite) cardinal $\kappa$ such that any open cover of $X$ has a subcover of cardinality ...