# Tagged Questions

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; ...

70 views

### Why does one necessarily need the triangle inequality

I'm studying basic topological, metric and normed spaces and I am curious why one of the axioms of both a metric and a norm is the triangle inequality. It makes some sense to me having the triangle ...
45 views

### Three Jordan curves made from paths

For continuous paths $\phi,\psi:[0,1]\to \mathbb{R}^2$ such that $\phi(1)=\psi(0)$, let $\phi*\psi$ denote the composition path and $\phi^{-1}$ the inverse path. Consider points $p,q\in\mathbb{R}^2$ ...
48 views

### metric space with no perfect set

Let $X$ be a complete separable metric space containing no perfect set of size greater than $1$. In other words every subset of $X$ has an isolated point. It is well known that $X$ must be countable....
35 views

### Surjectivity on the image of a annulus

I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself. ...
26 views

### Minimal prime ideals of the ring of continuous functions

Let $X$ be a topological space. Are there any conditions on $X$ which guarantee that that the minimal prime ideals of $C(X)$, the ring of real-valued continuous functions on $X$, have a nice ...
18 views

### Union of Interiors is Subset of Interior of Union

I'm teaching my self topology using a book I found. This is the forth part of a 4 part question. links to other parts: one, two, three . I'm trying to prove the following problem from a book I found: ...
39 views

### Closure of Union contains Union of Closures

I'm teaching my self topology using a book I found. This is the second part of a 4 part question. First part is here. I'm trying to prove the following problem from a book I found: Let $X$ be a ...
3k views

### If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?

If a nonempty set of real numbers is open and closed, $\mathbb{R}$ Why/Why not? In other words, are $\emptyset$ and $\mathbb{R}$ the only open and closed sets in $\mathbb{R}$? Why/Why not? I tried ...
17 views

### Intersection of Interiors contains Interior of Intersection

I'm teaching my self topology using a book I found. This is the third part of a 4 part question. links to other parts: one, two. I'm trying to prove the following problem from a book I found: Let ...
30 views

### Continuous function with real numbers and rational numbers…

I know if I consider real numbers with topology which generated by [a,b) there is a continuous function from R onto rational numbers with usual topology. Also, I know there is a continuous function ...
49 views

### Connectedness of suspension of a topological space

The suspension $\Sigma X$ of a topological space $X$ is defined as the quotient space $$\Sigma X=\dfrac {X\times \left[0,1\right]}{\sim}$$ where $(x,t)\sim (y,s)$ if and only if $s=t=0$, or $s=t=1$, ...
27 views

### Homotopy on a cylinder

Given a cylinder $C := \mathbb{R} \times S^1$, the fundamental group is $\pi_1 \cong \mathbb{Z}$. My basic question is: Why? I completely fail to see what the set of non-homotopic loops on the ...
56 views

### Is $Y$ homeomorphic to $\mathbb S^1$?

Let $Y = \mathbb S^1 \cup\{ (x,y): (x-2)^2 + y^2 =1 \}$ be a subspace of $\mathbb R^2$. Is $Y$ homeomorphic to $\mathbb S^1$? Is $Y$ homeomorphic to an interval? Can anybody please ...
25 views

### A bounded set in the plane

Consider the set in the plane described by the inequalities $x^2+3y\le e^y$, $y^2\le x+y$. Using Mathematica, I can see that the set is bounded. How can I prove that?
3k views

16 views

### embedding $T_0$ topology with specific properties in $(\Bbb{R},τ')^J$

Let $X$ be an infinite set and let $τ$ be a $T_0$-topology on $X$ that has no open finite subset and has no finite closed subset except empty set. For $x∈\Bbb{R}$ let $U_x=${$y∈\Bbb{R}:y<x$}, and ...
79 views

### Is the sphere with a diameter homotopy equivalent to a surface?

This is for a homework problem: Take the unit sphere $\mathbb{S}^2$ and join the north and south poles with a line segment. Is the resulting space homotopy equivalent to a surface? Intuitively, ...
702 views

### Why can we always take the zero section of a vector bundle?

$\require{AMScd}$ As I understand it, a rank $k$ vector bundle is a pair of topological spaces with a map between them $$E\xrightarrow{p}B$$ such that there exists an open cover $(U_\alpha)$ of $B$ ...
160 views

### Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
21 views

### Motivation for the definition of continuous maps on topological spaces [duplicate]

In any category where the objects are sets equipped with certain relations and operations, the notion of "morphism" arises perfectly naturally. (Generally, a morphism between objects is one that ...
39 views

4k views

53 views

### Find the limit point of set A in topology

Find limit points of set A A = {$n\sin\frac{1}{n}+(-1)^n\frac{1}{m}$ where $n,m\in N$} i think A' = {1, 0} but i can't prove this problem. Help
31 views

### Why is the empty set part of the topology in the following examples?

I am studying topology from Munkres. Why is the empty set an element of the topology that is schematically represented in the picture ?
### Show that the closure of $A$ is the intersection of all closed sets containing $A$, topology proof needed
I want to show that given $(X, \mathcal{T})$, we define $\overline A = \{x \in X| \forall U \in \mathcal{T}, x \in U \implies U \cap A \neq \varnothing\}$ (definition of closure from Munkres), then ...
### Does $\mu^*$ agree with $\mu$, measure space? [closed]
If $(X, \mathcal{A}, \mu)$ is a measure space, define$$\mu^*(A) = \inf\{\mu(B) : A \subset B,\,B \in \mathcal{A}\}$$for all subsets $A$ of $X$. I have a few questions? Is $\mu^*$ an outer measure? ...