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; ...

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17
votes
2answers
611 views

Topology on the space of paths

Let $X$ be a topological space, and define a path as a continuous map $\gamma : [a,b] \rightarrow X$. Two paths $\gamma : [a,b] \rightarrow X$ and $\phi : [c,d] \rightarrow X$ are equivalent ($\gamma ...
17
votes
3answers
687 views

Where do we need the axiom of choice in Riemannian geometry?

A friend of mine is a differential geometer, and keeps insisting that he doesn't need the axiom of choice for the things he does. I'm fairly certain that's not true, though I haven't dug into the ...
14
votes
1answer
122 views

a continuous bjiective map which is not a homeomorphism

Is there a bijective continuous function $f:\mathbb{Q}\rightarrow \mathbb{Q}$ that not a homeomorphism? I am not able to prove it or disprove it. The problem that the rationals is not even locally ...
13
votes
3answers
573 views

Fundamental Theorem of Algebra for fields other than $\Bbb{C}$, or how much does the Fundamental Theorem of Algebra depend on topology and analysis?

When proving the Fundamental Theorem of Algebra, we need to appeal to analytic and/or topological properties of $\Bbb{C}$ and $\Bbb{C}[z]$. Is this going to be necessary in general, and if so, to what ...
13
votes
2answers
591 views

If $f\circ g$ is continuous and $g$ is continuous what about $f$?

I don't know if $f$ is continuous. I believe that isn't necessarly continuous but I don't know some example. If it is continuous I don't know how to prove.
12
votes
1answer
387 views

Countable-infinity-to-one function

Are there continuous functions $f:I\to I$ such that $f^{-1}(\{x\})$ is countably infinite for every $x$? Here, $I=[0,1]$. The question "Infinity-to-one function" answers is similar but without the ...
12
votes
1answer
4k views

Prove $\epsilon$-$\delta$ definition of continuity implies the open set definition for real function

I need to prove that the $\epsilon$-$\delta$ definition of continuity implies the open set definition continuity for a real function. Here's my attempt. For any basis $V: (a, b)$ in the range, for ...
12
votes
1answer
589 views

Topological space that is not homeomorphic to the disjoint union of its connected components

As the title says, I'm looking for a counterexample to the statement that every topological space X is homeomorphic to the disjoint union of its connected components. I know that this is in fact ...
11
votes
3answers
1k views

Compactness of the Grassmannian

Let $V$ be a finite-dimensional inner product space. For $0 \leq d \leq \text{dim}(V)$, define the Grassmannian $G(V, d)$ to be the set of all $d$-dimensional linear subspaces of $V$, equipped with ...
10
votes
1answer
422 views

Infinity-to-one function

Are there continuous functions $f:I\to S^2$ such that $f^{-1}(\{x\})$ is infinite for every $x\in S^2$? Here, $I=[0,1]$ and $S^2$ is the unit sphere. I have no idea how to do this. Note: This is ...
10
votes
2answers
165 views

$|f(x)-f(y)|\le(x-y)^2$ without gaplessness

If $|f(x)-f(y)|\le(x-y)^2$ for all $x,y\in\mathbb R$, then it's easy to show that $f'=0$ everywhere, and the mean value theorem implies that that means $f$ is constant. If there were a gap in the ...
10
votes
1answer
546 views

How many points does Stone-Čech compactification add?

I would like to know how Stone-Čech compactification works with simple examples, like $(0,1)$, $\mathbb{R}$, and $B_r(0)$ (the open ball of $R^2)$. I've studied the one-point compactification and this ...
10
votes
2answers
858 views

Tychonoff Theorem in the box topology

A short question: Why does not the Tychonoff theorem (the arbitrary product of compact spaces is compact) hold in the box topology? I don't know how to show that there is no finite sub-cover of any ...
10
votes
2answers
1k views

Functions continuous in each variable

Suppose we have a map $f:X \times Y \rightarrow Z$, where $X,Y$, and $Z$ are topological spaces. Are there any conditions on $X$,$Y$, and $Z$ that would allow one to determine that $F$ is continuous ...
9
votes
2answers
1k views

Maximal ideals in $C(X)$ and Axiom of Choice

The following result are true if we assume full axiom of choice: A. If $X$ is a compact Hausdorff space, then every maximal ideal of the ring $C(X)$ has the form $A_p=\{f\in C(X); f(p)=0\}$. B. If ...
8
votes
1answer
124 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
8
votes
2answers
307 views

Ways to induce a topology on power set?

In this question, two potential topologies were proposed for the power set of a set $X$ with a topology $\mathcal T$: one comprised of all sets of subsets of $X$ whose union was $\mathcal T$-open, one ...
8
votes
2answers
298 views

Sufficient conditions for $M(X \times Y, Z)$ to be homeomorphic to $M(X, M(Y, Z))$

Let $Y^X$ denote the set of all functions $f: X \to Y$. If $X$ and $Y$ are topological spaces, let $M(X,Y)$ denote the set of all continuous maps $f: X \to Y$, endowed with the compact-open topology. ...
8
votes
3answers
455 views

Fiber bundle is compact if base and fiber are

I want to show that the total space $E$ is compact if the fiber $F$ and the base space $B$ are compact. Let $\pi$ denote the fiber projection. Since every point in $B$ has an open neighborhood $U$ ...
8
votes
1answer
3k views

About the interior of the union of two sets

Let $X$ be a topological space and let $A,B\subseteq X.$ We know that in general, we only have $$int(A)\cup int(B)\subseteq int(A\cup B). $$ My question is: When do we say that equality holds? I ...
8
votes
1answer
484 views

What are the requirements for separability inheritance

Suppose we have an arbitrary separable topological space $X$. What are some (possibly nonequivalent) minimal requirements to put on $X$ to ensure that every subspace of $X$ is separable? This is not ...
8
votes
2answers
1k views

Is the algebraic closure of a $p$-adic field complete

Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?) Why is (or why isn't) an algebraic closure $\overline{K}$ complete? Maybe this holds more ...
8
votes
1answer
147 views

Lindelöf'ize a space?

Although much weaker, Lindelöf is in the same spirit of compactness. It occurs to me whether there is such a thing like "Lindelöf'ization", since there are various kinds of compactification process ...
8
votes
2answers
442 views

Quotient map from $\mathbb R^2$ to $\mathbb R$ with cofinite topology

Let $X = \mathbb R$ under the cofinite topology. Is there a quotient map $q : \mathbb R^2 \rightarrow X$? Intuitively, this seems like it should be false, since $\mathbb R^2$ has "too many" open sets. ...
8
votes
1answer
457 views

Nice underestimated elementary topology problem

There is a nice elementary topology problem (proposition) that is often missing from the introductory books on the topic. PROBLEM. Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a ...
8
votes
3answers
999 views

Example of Hausdorff space $X$ s.t. $C_b(X)$ does not separate points?

We know the Stone-Weierstrass theorem for locally compact Hausdorff spaces (LCH) which states the following: Theorem: Suppose $X$ is LCH. A subalgebra $\mathcal{A}$ of $C_0(X)$ is dense if and ...
7
votes
2answers
85 views

For which matrices $A \in \mathscr{M}_n(\mathbb C)$ is the similarity class of $A$ closed?

What are the matrices $A \in \mathscr{M}_n(\mathbb C)$ for which the similarity class is closed? What about the same question if we replace $\mathbb C$ by $\mathbb R$?
7
votes
2answers
219 views

Every preorder is a topological space

Let $(X,\leq)$ be a preorder, i.e. $\leq$ is a reflexive and transitive binary relation on $X$. I want to show that $\leq$ induces a topological structure on $X$. Hence I need to specify when a subset ...
7
votes
1answer
177 views

Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$?

Is $\def\Aut{\operatorname{Aut}}\Aut(\mathbb{I})$ isomorphic to $\Aut(\mathbb{I}^2)$ ? ($\mathbb{I},\mathbb{I}^2$ have their usual meaning as objects in $\mathsf{Top}$). I show some of one of my ...
7
votes
2answers
260 views

A subset of a metric space is closed iff its intersection with every compact subset is closed [closed]

I want to show that a subset of a metric space $X$ is closed iff its intersection with every compact subset of $X$ is closed
7
votes
5answers
1k views

Motivation for the importance of topology

Starting from tomorrow, I will be tutoring some undergraduate students following a course in general topology. I am looking for examples motivating the importance of topology in mathematics which can ...
7
votes
2answers
306 views

how to prove that sigma-compact space is D-space

Please, help me. In the paper "A survey of D-spaces" by Gary Gruenhage it is written that it is easily seen that $\sigma$-compact spaces are D-spaces. Unfortunately, I don't know how to show it. ...
7
votes
2answers
826 views

Graph of a function homeomorphic to a space implies continuity of the map?

if $X,Y$ are topological spaces and $f: X \rightarrow Y$ is a continuous map then it can be shown that the graph of $f$, $G_{f}$, is homeomorphic to $X$. But is the converse true? that is: if $G_{f} ...
6
votes
1answer
301 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 ...
6
votes
1answer
118 views

Is there a non-locally compact Hausdorff space in which all infinite compact sets (of which there is at least one) have nonempty interior?

Here is the background material from which I am working: The Cantor set is an uncountable compact Hausdorff space with empty interior. In a locally compact Hausdorff space, each countable set has ...
6
votes
2answers
634 views

Product of two compact spaces is compact

I read the proof that uses tube lemma and I do not have any problem with it but I cannot see what is wrong with the proof that first came to my mind: Let $X$ and $Y$ be compact spaces. Let ...
6
votes
1answer
502 views

A question about the contractibility of the Sierpinski space

The two-point Sierpinski space is usually defined as follows: Let $X =\{x,y\}$ be the two-point space where the only open sets are $X, \varnothing, \{x\}$. I think from this it can be inferred that ...
5
votes
1answer
85 views

Example of non-homeomorphic compact spaces $K_1$ and $K_2$ such that $K_1\oplus K_1$ is homeomorphic to $K_2\oplus K_2$

Once I heard that there exists two compact spaces $K_1$ and $K_2$ which are non-homeomorphic, but with $K_1\oplus K_1$ homeomorphic to $K_2\oplus K_2$ (where $\oplus$ denotes the topological sum). Is ...
5
votes
0answers
161 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
5
votes
3answers
218 views

Construct a complete metric on $(0,1)$

Can anyone construct a complete metric on $(0,1)$ which induces the usual subspace topology on $(0,1)$ ?
5
votes
2answers
2k views

connected sum of torus with projective plane

I would like to understand how to prove that the connected sum $\mathbb{R}P^2 \# T^2$ of the projective plane with a torus is homeomoprhic to $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2$. I got ...
5
votes
4answers
513 views

How to show that continuous functions between metric spaces agree on a closed set

Let $(X,d)$ and $(Y,d')$ be metric spaces, and let $D$ be a dense subset of $X$. Show that: If $f:X\to Y$ and $g:X\to Y$ be continuous, then the set $\{x\in X\mid f(x)=g(x)\}$ is closed.
5
votes
1answer
296 views

An exercise about finite intersection property in $T_1$ space

Let $X$ be a $T_1$ space. Let $\mathfrak {D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. Show that there is at most one point belonging to ...
5
votes
0answers
109 views

What is special about simplices, circles, paths and cubes?

There are some ubiquitous families of graphs — the complete graphs (or simplices) $K_n$, the circle graphs $C_n$, the path graphs $P_n$, and the hypercube graphs $Q_n$ — that intuitively ...
5
votes
2answers
554 views

In set theory, what does the symbol $\mathfrak d$ mean?

What's meaning of this symbol in set theory as following, which seems like $b$? I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma? Thanks ...
5
votes
4answers
7k views

If A is a subset of B, then the closure of A is contained in the closure of B.

I'm trying to prove something here which isn't necessarily hard, but I believe it to be somewhat tricky. I've looked online for the proofs, but some of them don't seem 'strong' enough for me or that ...
5
votes
1answer
2k views

When is the union of topologies a topology?

The union of two topologies on some set may or may not be a topology. When is it a topology?
5
votes
2answers
591 views

Analytic Applications of Stone-Čech compactification

Following Bredon's Topology and Geometry, we let $\mathcal{F}$ be the set of all continuous maps $f:X \to [0,1]$ on a completely regular space $X$, define $X \xrightarrow{\Phi} [0,1]^{\mathcal{F}}$ by ...
4
votes
1answer
103 views

universal property in quotient topology

The following is a theorem in topology: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/\sim$ be the canonical projection. If $g : X → Z$ is a continuous ...
4
votes
2answers
285 views

Proof of $(0,1)$ is not compact with usual metric.

In the proof we say $\left\{\left(\frac1n,1\right):n\geq 1\right\}$ is an infinite cover with no finite subcover. But, $(0,1)$ set also belongs to cover mentioned above. We can say $\{(0,1)\}$ is a ...