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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2
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1answer
118 views

Family of continuous maps generates the topology of X?

I want to know if this is true: Let $X$ be a Tychonoff space and let $\{f_i:i\in I \}$ be a family of continuous maps $f:X\to [0,1]$ such that it separates points and closed sets. Then $\{f_i:i\in I ...
1
vote
1answer
59 views

Extending the function in $\Bbb R^n$

Let's take a function $f:\Bbb R^k \supset A \rightarrow \Bbb R^m$, such that: $(1): \forall_{x,y\in A}:|f(x)-f(y)| \le |x-y|$. Is it true that every such function can be extended to function $f':\Bbb ...
5
votes
2answers
417 views

Question on problem: Equivalence of two metrics $\iff$ same convergent sequences

Community! Im working on the following problem: Let $X$ be a non-epmty set and $d_1,d_2$ metrics on X. Show that the following conditions are equivalent: 1) $d_1$ and $d_2$ are equivalent, i.e. ...
1
vote
0answers
143 views

The Lebesgue number property and uniform continuity (proof check)

Theorem If $f$ is continuous on a compact metric space $X$, then $f$ is uniformly continuous on $X$. Proof Let $\epsilon>0$. For any $y\in X$ there is a $\delta_y$ such that $d(x,y)<\delta_y$ ...
1
vote
2answers
177 views

Is this set closed? (Finding the limit points of a set)

Prove that this set is closed: $$ \left\{ \left( (x, y) \right) : \Re^2 : \sin(x^2 + 4xy) = x + \cos y \right\} \in (\Re^2, d_{\Re^2}) $$ I've missed a few days in class, and have apparently ...
18
votes
10answers
5k views

How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
2
votes
2answers
98 views

Proving the graph of an equivalence relation is closed in a product space

Let $X = \mathbb R^n \setminus \{0\}$, $n\ge2$. Let $\sim$ be the equivalence relation defined by $x\sim y$ iff there exists a non-zero real number $k$ such that $kx=y$. Prove that the graph of ...
3
votes
2answers
376 views

Regular Borel Measures equivalent definition

Please help me understand how the below definition is equivalent to the standard definition of regularity which says "that a measure is regular if for which every measurable set can be approximated ...
3
votes
2answers
93 views

Is every countably compact space feebly compact?

A topological space is said to be feebly compact if every locally finite cover by nonempty open sets is finite. Every compact space is feebly compact but how about countably compact spaces?
5
votes
1answer
235 views

Under what condition only does every compact subset of $X$ is closed implies $X$ Hausdorff?

It is trivial to see that: If $X$ is Hausdorff, then every compact subset of $X$ is closed. I am asking under what condition does the converse hold, i.e. when does If every compact subset of $X$ is ...
1
vote
1answer
64 views

An example for a non-precompact minimal topological group.

Do you have an example of a non-precompact minimal topological group? A topological group $(G,\mathcal T)$ is said to be minimal iff it is Hausdorff and for any compatible Hausdorff topology ...
1
vote
1answer
166 views

There exists a continuous function that satisfies this property

Let $X$ be a non-compact subset of $\mathbb{R}$. I want to show that there a continuous function $f: X \to \mathbb{R}$ such that $f$ is bounded but does not attain its bounds. I think that there ...
3
votes
2answers
43 views

A question on the linear order space

How to see any linear order space (LOTS) is regular? In other words, is it always regular? Thanks for your help. Any help will be appreciated.
9
votes
1answer
2k views

Why is the graph of a continuous function to a Hausdorff space closed?

Say I have two topological spaces given by $(X,\mathscr{T}_X)$ and $(Y,\mathscr{T}_Y)$ where $Y$ is Hausdorff. In addition say I have a function $f:X\rightarrow Y$, and let it be continuous. I want to ...
1
vote
2answers
194 views

Density of $\mathbb{Q}^n$ in $\mathbb{R}^n$

$\mathbb{Q}$ is dense in $\mathbb{R}$ (with the standard topology). I'm pretty sure that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$ too. Is there an easy argument to prove that without reproducing the ...
7
votes
1answer
408 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
4
votes
2answers
51 views

Closedness with respect to an open cover

This is something that came up when I was studying something else, but I am wondering whether the following topological fact is true. Let $X$ be a topological space, and $\{U_i\}_{i=1}^n$ a ...
2
votes
1answer
120 views

Is an ultra-net a subnet of all it's subnets?

Let $(x_d)_{d\in D}$ be a net on a set $X$ and the set $$\{\{x_d\mid d\ge p\}\mid p\in D \}$$ be a base for an ultrafilter on $X$. Let $(x_{d'})_{d'\in D'}$ be a subnet of $(x_d)_{d\in D}$. Is ...
7
votes
2answers
466 views

If the graph $G(f)$ of $f : [a, b] \rightarrow \mathbb{R}$ is path-connected, then $f$ is continuous.

Yesterday I woke up thinking about this question, and I believe I have a proof, but I'm not sure of its validity. Let $\gamma : [a, b] \rightarrow \mathbb{R}^{2}$ be a path from $(a, f(a))$ to $(b, ...
1
vote
2answers
143 views

Necessary and Sufficient Condition for two metrics to have same open sets.

There are couple of independent conditions like one being scalar multiple of another, or if $$d_p(x,y)=(x^p+y^p)^{1/p}$$ then all $d_ps$ and $d_qs.$ which guarantee that open sets are same under these ...
2
votes
1answer
41 views

Analogous notion of knot complements for braids

Knots/links seem to be studied quite a lot for their topological connection to 3-manifolds by considering knot complements in $S^{3}$. Is there an analogous topological entity for braids? They appear ...
7
votes
1answer
528 views

Contractibility of convex set

Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$. Let $Q_m\subset \Omega$ denote ...
2
votes
1answer
50 views

Proving the closedness of given sets

I want to show that the sets $\{(x, y):xy = 1\} $ and $\{(x, y):x^2+y^2 = 1 \} $ are closed in $\mathbb{R}^2$. Geometrically, it is clear that both sets contains all of its limit points in ...
6
votes
2answers
309 views

Understanding quotient topology

Going through some wiki notes and books I found that a quotient space (also called an identification space) is, the result of identifying or "gluing together" certain points of a given space. The ...
1
vote
1answer
253 views

topology in R infinity

What does the following sentence mean and why is that true: "The nonnegative orthant in $R^{\infty}$ has empty interior in product topology" Thank you!
5
votes
1answer
88 views

$\kappa\psi (x,X)\leq \psi (x,X)$

The $\kappa$-pseudocharacter $\kappa\psi (x,X)$ of a space $X$ at a point $x\in X$ is the smallest infinite cardinal number $\tau$ such that there exists a family $\gamma$ of $\kappa$-sets in $X$ ...
2
votes
1answer
65 views

A closed form for a particular topology.

I am trying to find some sort of 'closed form' (if possible) of a particular topology generated by the sets: $({x\in\mathbb{R}\ \vert x\geq a}), a\in \mathbb {R}$. Thanks !
2
votes
3answers
2k views

Prove a square is homeomorphic to a circle

$s:=\{|x|\le 1,|y|\le 1\} $ $c:=\{{x}^{2}+{y}^{2}\le1\}$ Prove $\overset{\circ}{s} \cong \overset{\circ}{c}$ ok... not to sure what to do. I think $\overset{\circ}{s} ...
0
votes
1answer
52 views

Codimensionality: On Cardinality of Linear Equations

How does the codimension of a subspace give the number of linear equations needed to define the subspace?
3
votes
2answers
105 views

Every set of non-overlapping disks in the plane is at most countable?

In the Euclidean plane $\mathbb{R^2}$, the set of points inside a circle is a disk. Can we claim that every set of non-overlapping disks in the plane is at most countable? My intuition says it must ...
0
votes
1answer
77 views

Study the continuity of $f: (\mathbb{Z},\tau_i) \to (\mathbb{R}_+,\tau)$

Let $\mathbb{R}_+:=[0,\infty)$ $$ f:\mathbb{Z} \to \mathbb{R}_+$$ $$ n\mapsto f(n):={n}^{2}$$ and $\tau$,$\tau_1$,$\tau_2$ be the topologies defined as follows: $$\tau :=\{\emptyset,\mathbb{R}_+ ...
0
votes
2answers
98 views

continuous function from $[\pi,2\pi]\to \mathbb{R}^2$

$f:I=[\pi,2\pi]\to \mathbb{R}^2$ be given by $f(t)=(\cos t,\sin t)$ which of the following are necessarily correct? $1$. $\exists t_0\in I$ such that $f'(t_0)=\frac{1}{\pi}(f(2\pi)-f(\pi))$ $2$. ...
0
votes
2answers
73 views

Prove the following for the relation of equivalence.

We define the following equivalence relation on ${\mathbb{R}}^{2}$ $$(x,y) \sim (X,Y) $$ if and only if y=Y and $$(x,y) \Join (X,Y) $$ if and only if $${x}^{2} + {y}^{2}= {X}^{2}+{Y}^{2}$$ Prove ...
0
votes
2answers
121 views

Closed and open sets

By regarding the real numbers with their natural topology, my textbook says, that: $$ \left\{2 \pi n+\frac{1}{n}\;\bigg|\;n \in \mathbb{N} \right\}$$ is closed, which i understand, as every sequence ...
0
votes
1answer
84 views

derivative of a function with disconnected domain

$f:A\cup E\to\mathbb{R}^2$ be differentiable, where $$A=\{(x,y)\in\mathbb{R}^2:\frac{1}{2}<x^2+y^2<1\}$$ and $$E=\{(x,y)\in\mathbb{R}^2:(x-2)^2+(y-2)^2<\frac{1}{2}$$ which of the following ...
4
votes
3answers
868 views

$\mathbb{Q}$ is not open, is not closed, but is the countable union of closed sets.

I want to prove that $\mathbb{Q}$ (the set of rational numbers) is not open, is not closed, but is the countable union of closed sets. I tried to show that $\mathbb{Q}$ doesn't contain all of its ...
1
vote
1answer
189 views

Qualifying Exam Question on Manifolds

I am practicing qualifying exam problems and I am having trouble with the following question. Any help is greatly appreciated. Let $P$ be a polygon with an even number of sides. Suppose that the ...
1
vote
1answer
104 views

How to prove that every space can be decomposed into scattered and perfect subspace

How to prove the following claim: A space $X$ can be represented as the union of two disjoint subset $A$ and $B$, where $A$ is a scattered and $B$ satisfies $B=B'=:\{x: x \text{ is the accumulation ...
1
vote
2answers
106 views

On the small first countable regular spaces

Here are some definitions. A countably infinite closed discrete set $A \subset X$ has property $D$ in $X$ provided there exists a discrete family of open sets $\{U_a : a \in A\}$ ...
2
votes
1answer
63 views

Proving the existence of a closed set

Let $X$ be a compact space and $f: X \to X$ be a continuous map. Show that there is a non-empty closed set $E \subseteq X$ such that $E = f(E)$. First, something is a bit fishy about this ...
2
votes
1answer
55 views

Tychonoff spaces are perfect images of extremally disconnected spaces.

Encyclopaedia of Mathematics, p524, says that Completely regular spaces are perfect images of extremally disconnected spaces. But no process and no reference has been presented to prove. Any ...
0
votes
1answer
101 views

Prove a subbase for this topology

Let $X$ be a set. We define the following topology on $X$: $$\tau=\{S\subseteq X:X\setminus S\text{ is finite}\} \cup \{0\}$$ Prove $\{X\setminus\{x\}:x \in X\}$ is a subbase of this topology.
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2answers
101 views

Does there exist a retraction for these spaces?

I have some qualifying exam questions using retractions that I do not know how to solve. The only tool that I know for solving retraction problems is by using the fundamental group trick. Here are ...
1
vote
2answers
137 views

Prove all neighbourhoods are open in this topology

Let $X$ be a set. We define the following topology on $X$: $$\tau=\{S\subseteq X:X\setminus S\text{ is finite}\}$$ How do you prove all neighbourhoods are open in this topology. Thanks
1
vote
1answer
177 views

Is $d'$ a distance? Does it define the same topology on $M$ as $d$?

Let $(M,d)$ be a metric space. Define the following function $d'(x,y):=\dfrac{d(x,y)}{d(x,y)+1}$ for every $x,y$ in $M$. Is $d'$ a distance? Does it define the same topology on $M$ as $d$? I didn't ...
1
vote
2answers
163 views

Hausdorff space and Cantor's intersection theorem

$X$ is a Hausdorff space, $C_i$ is a non-empty closed subset of $X$ and $C_{k+1}\subseteq C_k$ , show that $\displaystyle \bigcap_{i\in \mathbb{N}} C_i$ is compact. I tried to prove by ...
0
votes
1answer
59 views

A subset of $\mathbb{C}^{n+1}$

I have the following topology question. Let $U$ be an open subset of $\mathbb{C}^n$. We consider the subset $U_1$ of $\mathbb C^{n+1} \colon U_1=\{a(1,z):a \in \mathbb{C}−\{0\},z \in U\}$. Could ...
1
vote
1answer
58 views

Extension on pseudometric spaces using uniform continuity

I would like to know if the extension theorem of uniformly continuous functions can be generalized to pseudometric spaces. That is, let $X,Y$ be a pseudometric spaces and $D\subset X$ a dense ...
3
votes
3answers
94 views

An equivalent definition of open functions

In any topological space: How can I prove That $f$ is open if the pre-image of $\operatorname{cl}(A)$ is contained in the closure of pre-image of $A$, Can some one give me a hint. Actually, the ...
3
votes
3answers
949 views

Show that a function $f$ is strictly increasing on the interval $[0,1]$

Let $f:[0,1]\to\mathbb{R}$ be a continuous function that does not take on any of its values twice and with $f(0)<f(1)$. Show that $f$ is strictly increasing on $[0,1]$.