0
votes
2answers
41 views

Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
1
vote
2answers
21 views

Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
1
vote
1answer
36 views

What is the definition of Lindelöf space?

My definition for "countable set" is a set with the cardinal $\aleph_0$ and "at most countable set" is a set $A$ such that $|A|≦\aleph_0$. Till now, my definition for Lindelöf space is a topological ...
0
votes
0answers
60 views

Confused by definition of an open set in “All the Mathematics You Missed”

On page 66 of Thomas Garrity's "All the Mathematics You Missed", Garrity gives the following definition of an open set in $\mathbb{R}^n$: A set $U$ in $\mathbb{R}^n$ will be open if given any $a ...
0
votes
1answer
42 views

Is there a name for a property defined in terms of open sets?

We know that if a property is defined in terms of open sets then the property is preserved under a homeomorphism. Is there a name for such a property?
0
votes
0answers
15 views

Definition of Non-characteristic Manifold

What is the definition of non-characteristic manifold in the following context. "Since $ f (x, y)=0 $ is assumed to be a non-characteristic singularity manifold, we have $ f_{x}\neq 0 $." Thanks, ...
2
votes
1answer
62 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The ...
0
votes
1answer
49 views

Definition singular manifold

I'm looking for the definition of a singular manifold. I haven't found it yet. For instance, in $\mathbb{R}^4$, with $f(x,y,z,t)=xy-zt$, $f^{-1}(0)$ is a singular submanifold. I only found a few ...
0
votes
1answer
68 views

Uniform Space: Neighborhood System [closed]

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! As the idea behind uniform spaces is to represent a ...
5
votes
1answer
223 views

Topology: Opens vs Neighborhoods

Disclaimer: This thread is meant informative and therefore written in Q&A style. The problems are highlighted in bold face. The axiomatization of topology can be done in various ways all of ...
1
vote
1answer
40 views

definition of limit of function on topological spaces

Def.: let be $(A,\tau)$,$(C,\zeta)$ two topological spaces, $f \in C^E$, with $E \subseteq A$, and $x_0$ an accumulation point of $E$, a point $l \in C$ is limit of $f$ as $x$ approaches $x_0$ if ...
1
vote
3answers
85 views

Alternative to epsilon-delta definition

Back in Multivariable Calculus, I remember my professor, when explaining why he decided to skip a rigorous definition of limit (the reason was that those of us that would continue in math would go ...
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!
0
votes
2answers
25 views

Formalize definition of subbase of a topology

Def.: let be $(A,B)$ a topological space, and $C \subseteq B$, "$C$ is subbasis of $B$ if $$\{X|\exists X_1,X_2,...,X_n \in C(X=\bigcap_{i=1}^n X_i)\} \text{ is basis of } B$$ Is it correct?
1
vote
1answer
87 views

Topology: Interior

The neighborhood filters satisfy: $$\forall N\in\mathcal{N}(x):\qquad x\in N$$ $$\forall N\in\mathcal{N}(x)\exists M_0\in\mathcal{N}(x):\qquad N\in\mathcal{N}(m)\text{ for all }m\in M_0$$ Define the ...
2
votes
1answer
57 views

formalize definition of topology

In my studies I used this definition of topology, but I am reading on wikipedia a different definition... I thought to formalize: Def. let be $A$ a set and $B \in \mathcal{P}(\mathcal{P}(A))$, ...
1
vote
1answer
39 views

A special kind of metric-spaces

Is there a special name for those metric-spaces or topological spaces in which every non-empty open set is uncountable ?
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 $r>0$, $\overline{B}(x;r)$ ...
5
votes
3answers
100 views

Why must an interior point of $E$ be an element of $E$?

This question takes place in a general metric space $X$. Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$. This is like the ...
0
votes
0answers
79 views

Is it false that the complement of an open set is closed?

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a continuous function. Let $Z(f)$ be the zero of $f$. Prove that $Z(f)$ is closed. This is one of problems in my mid-term exam. I have used ...
0
votes
1answer
29 views

Convergence of vector spaces

I'm considering vector spaces over $\mathbb{R}$ or $\mathbb{C}$ : $(E_n)_n$ is a sequence of vector space (all included in $\mathbb{R}^p$ for example, with $p$ fixed). What meaning(s) do we usually ...
1
vote
1answer
49 views

Some troubles about topology and definition of a Vector Bundle

Disclaimer: It's heavily related to my old question : Visualizing the Topology of a Vector Bundle but I wanted to open a new question because the former had already got an answer and this time my ...
0
votes
1answer
19 views

Formal notation for finite intersection

How to state following sentence formally? Let $\tau$ be family of sets (possibly infinite). Any finite intersection of sets in $\tau$ is in $\tau$. My attempt is: $\tau$ is family of sets and ...
1
vote
1answer
43 views

“internal” definition of complete regularity?

There is something strange (I think) about the complete regularity separation axiom. Consider the definitions. T0 means for every two distinct points there is an open set containing exactly one of ...
6
votes
2answers
214 views

The phrases “has … in ” vs. “contains … of” in Baby Rudin

Consider the following two statements. (Assume $E \subseteq K$.) $E$ has a limit point in $K$. vs. $E$ contains a limit point of $K$. What do they each mean and how are they different?
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. ...
3
votes
1answer
47 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
5
votes
4answers
265 views

What is the *exact* definition of a bounded subset in a metric space (in relation with the Heine-Borel Theorem)?

I see quite a lot of different definitions of a bounded space. For instance, from nLab: Let $E$ be a metric space. A subset $B⊆E$ is bounded if there is some real number $r$ such that ...
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
2answers
41 views

Point as an element of an affine space vs point as an element of a topological space?

I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong ...
0
votes
2answers
48 views

Is this predicate expressing compactness?

I don't quite know why definitions of "compact" use the expression "arbitrary collection". Am I correct in thinking that the following predicate is a definition of compact? Let $X$ by a set with ...
3
votes
3answers
136 views

Definition of Door Space

Every reference I can find regarding (topological) door spaces gives the following definition almost verbatim: A door space is one in which every subset is either open or closed. [emphasis mine] ...
0
votes
1answer
74 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
0
votes
1answer
76 views

Beginnings of Topology: Homeomorphisms

Why is a knot and a circle homeomorphic? The general definition of a homeomorphism requires that you be able to deform each to one another.
0
votes
1answer
60 views

Difference between the concepts of graph and trace

I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ...
1
vote
2answers
26 views

A confusion regarding the definition of continuous functions.

Let us suppose $f:X→Y$ is a continuous, non-surjective mapping. Also assume that there is an open set $A\subset Y$ which contains points which are not in $f(X)$. What would $f^{−1}(A)$ be? Would it ...
0
votes
2answers
95 views

definition of separation axioms in topology

I am learning the Separation Axioms and came across the definition of regular space. In the definition they say, "Suppose the one point sets are closed in $X$" My question is: how can one point sets ...
3
votes
2answers
113 views

What is compactification generally?

In wikipedia, compactification is defined as an topological imbedding $f:X\rightarrow Y$ such that $f(X)$ is dense in $Y$. However, Munkres-Topology requires $Y$ to be Hausdorff to be called a ...
0
votes
2answers
52 views

Confusion about the definition of a “loop” in a topological space

A continuous map $f:[0,1] \to X$ is called a path and if $f(0)=f(1)$ then it is called a loop. But any loop looks like a circle which is not a function as it is not well defined. How did it ...
1
vote
0answers
76 views

About definition of topology

let be $m$ a function "$m: X \to \mathcal{P}(\mathcal{P}(X))$" (and I denote: $m(x) := m_x, \forall x \in X $), $m$ is topology on $X$ if: 2)$ \forall x \in X (\forall t \in m_x(x \in t)) $ 3)$ ...
2
votes
2answers
58 views

Definition verification from two different books?

In Kaplansky's Set Theory And Metric Spaces, he mentions a useful example of a neighborhood of $x$ is a closed ball with center $x$. However, one of the theorems in baby Rudin is "Every neighborhood ...
0
votes
3answers
39 views

Topology generation

What does it mean for a topology to be generated? For example $X=\mathbb{R}$ be topology generated by $[a,b)$. Isn't the topology a collection of open sets? $[a,b)$ is not open though.
1
vote
1answer
88 views

A function “extends” to the cone on X

I have the following statement: A map $f : X \rightarrow Y$ is nullhomotopic if and only if it extends to the cone on $X.$ My problem is that I have no idea what "extends" means in this statement (I ...
1
vote
1answer
81 views

If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous?

If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous? I'm looking at a proof where they only show that $f$ is continuous and 1-1. Then I looked ...
0
votes
1answer
70 views

Definition of Dedekind cut as initial segment

the definition of Dedekind cut by initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B \neq \emptyset$, $B$ is Dedekind cut of $A$ under $\preceq$ ...
0
votes
1answer
82 views

Definition of initial segment

the definition of initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall b \in ...
1
vote
1answer
61 views

Is it true that the definition of an open subset in a metric space is different from the combination of the definitions of subsets and opens sets?

Dear reader of this post, I have a question concerning the equivalence of two definitions of open subsets (in metric spaces). To avoid confusion, I will state the two definitions and then ask my ...
5
votes
2answers
152 views

Why is a section of a sheaf over closed set defined this way?

Why is a section of a sheaf $F$ over closed set $S \subset X$ is defined as inductive limit $$ \varinjlim_{S\subset U} F(U)\; ?$$ From my point of view, we should define it as a function, which each ...
2
votes
1answer
89 views

Definition of the higher dimensional mapping tori

This is proving harder to search for than I imagined. The usual definition of a mapping torus $\mathcal{M}_h$ associated to a homeomorphism $h\colon X\rightarrow X$ on a topological space $X$ is the ...
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 ...