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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4
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3answers
58 views

Prove that if $A$ is both open and closed, $A=\mathbb R$.

Suppose $A$ is a non-empty subset of $\mathbb R$. Prove that if $A$ is both open and closed, $A=\mathbb R$. I think I'm supposed to assume that $A$ is not equal to $\mathbb R$ and derive a ...
0
votes
0answers
12 views

Necessary and sufficient condition for an orthonormal system to be total

Let $H$ be a Hilbert space over a field $\mathbb K$. Prove that an orthonormal system $\{a_n\}_{n=1}^{\infty}$ in $H$ is total if and only if: $\forall$ $x \in H$, the following holds: ...
2
votes
1answer
17 views

Image of continuous injective map has empty interior.

Let $\varphi :\left [ 0,1 \right ]\rightarrow \mathbb{R^2}$ be a continuous injective map. Let $I = \varphi \left ( \left [ 0,1 \right ] \right )$ be the image of this map. Prove that $I$ has empty ...
1
vote
1answer
21 views

Dense sets and Empty Interior

if $A$ is dense in $X$, is there a relation which shows in which cases $A$ has empty interior ? $\mathbb{Q}$ has an empty interior as a dense set in $\mathbb{R}$, so does its complementary in ...
1
vote
2answers
19 views

Some question about orthogonal complement

Let $H$ be a Hilbert space and $Y$ is a closed subspace we denote $Z$ for a orthogonal complement of $Y$ How can I prove that $Z$ is a closed subspace of $X$ (I want to prove subspace and closed ) ...
0
votes
0answers
17 views

A question on product space [duplicate]

If $|X|=\mathfrak c$, then what is the cardinality of the product space $X^{\omega}$? Thanks very much.
2
votes
1answer
25 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
0
votes
1answer
21 views

Example 4, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How to figure out saturated open sets?

Let $X$ be the closed unit ball $$ \{ \ x \times y \ \colon \ x^2 + y^2 \leq 1 \ \} $$ in $\mathbb{R}^2$, and let $X^*$ be the partition of $X$ consisting of all the one-point sets $\{ \ x \times y ...
1
vote
1answer
35 views

Find two disjoint open sets $U, V$ such that $A\subseteq U, B\subseteq V$ where $A,B$ are closed.

Let $A, B$ be two disjoint closed subsets of a certain metric space $(M,d)$. Show that there exist disjoint open subsets $U, V \subseteq M$ such that $A\subseteq U, B\subseteq V$. Give ...
0
votes
1answer
24 views

Prob. 3, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How is this map a quotient map that is neither open nor closed?

Let $\pi_1 \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be projection on the first coordinate. Let $A$ be the subspace of $\mathbb{R} \times \mathbb{R}$ consisting of all points $x \times y$ ...
0
votes
1answer
26 views

I think $A,B$ must be closed and disjoint

Prove that in every metric space, $(X; d)$, is possible find a continous function$f\colon X\to \mathbb{R}$ such, if $ A $ and $ B $ are two subsets of $ X $ then $ f(x) = 1 $, for every $ x\in A $ ...
1
vote
0answers
14 views

Prove that the “additive” operation of the module($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=$($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar ...
2
votes
1answer
11 views

Hausdorff spaces for continuous bijections

I have the following question being posed: Suppose $f:X\rightarrow Y$ is a continuous bijection. Prove that if $Y$ is Hausdorff, then $X$ is also Hausdorff. Here's my attempt: Consider any $a,b\in ...
0
votes
1answer
23 views

Find a suitable counterexample?

Is the following statement true or false? If a sequence $(x_n)$ with an infinite range $\{ x_n : n \in \mathbb{N} \}$ has precisely one accumulation point, then $(x_n)$ converges. I know the ...
1
vote
3answers
27 views

Show that the closed ball is closed in $\mathbb{R}^p$

Let $r>0, p \in \mathbb{N}$ be given. Show in detail that the closed ball $\{ x \in \mathbb{R}^p : ||x|| \leq r \}$ is closed in $\mathbb{R}^p$. Let $A = \{ x \in \mathbb{R}^p : ||x|| \leq r ...
-8
votes
0answers
43 views

Id proof this quastion [on hold]

Id proof this quastion pleace tomorw to me
2
votes
2answers
22 views

Examples of path component maps

I understand what needs to be done for the first part, i have to somehow map $1$ point onto $1$ point, in a map where there exists $2$ points... so the inverse map is injective, but how is this ...
4
votes
3answers
76 views

Mistake in (Baby) Do Carmo? Elementary topology of surfaces.

If you have the book, it's proposition 2 of section 5.3. If not, the proposition reads: Given any two points p and q $\in$ a regular, connected surface S, there exists a parameterized piecewise ...
2
votes
0answers
15 views

$\operatorname{Fr}( p(\overline U) )$ where $p : X \to Y$ is a closed, not necessarily continuous, surjection, and $U \subset X$ is open

Question: Let $p : X \rightarrow Y$ be a closed (not necessarily continuous) surjection. If $U$ is open, then $$\operatorname{Fr} ( p(\overline U) ) \subset p(\overline U ) \cap p(X - U).$$ I ...
2
votes
2answers
53 views

Example 1, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How to verify that this map is closed?

Let $X$ be the subspace $[0,1] \cup [2,3]$ of $\mathbb{R}$, and let $Y$ be the subspace $[0,2]$ of $\mathbb{R}$. The map $p \colon X \to Y$ defined by $$ p(x) \colon= \begin{cases} x \ &\mbox{ ...
1
vote
0answers
17 views

Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
1
vote
0answers
12 views

Finite intersection of arbitrary union not stable for arbitrary unions

It is a set-theoretic exercise to prove that the set of arbitrary unions of finite intersections of sets is still stable under finite intersections. However it is not true that finite intersection of ...
1
vote
1answer
20 views

A proof related to diameter of a simplex S

Question: Prove that the diameter $\mathcal p(S)$ of a simplex $\mathcal S$ equals the greatest Eucledian distance between two vectors in the simplex. My opinion: We all know what every vector in the ...
1
vote
0answers
11 views

Existence of real valued function continuous at $\mathbb Q$ discontinuous at $\mathbb R\backslash \mathbb Q$ [duplicate]

Does there exist a real-valued function of a real variable which is continuous at every rational point and discontinuous at every irrational point?
2
votes
1answer
61 views

Section 22 in Munkres' TOPOLOGY, 2nd edition: How to establish this equivalence?

Let $X$ and $Y$ be topological spaces; let $p \colon X \to Y$ be a surjective map. Then $p$ is said to be a quotient map provided a subset $U$ of $Y$ is open in $Y$ if and only if $p^{-1}(U)$ is ...
0
votes
0answers
25 views

Prob. 4, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to decide which cases to consider?

We need to show that the ordered square satisfies the first countability axiom. I'm not able to decide as to which separate cases to consider. By definition the ordered square is the product $I ...
0
votes
1answer
37 views

Is a limit point compact subspace of a Hausdorff space necessarily closed? [on hold]

I think the answer should be "no", but I can't give a counter-example.
0
votes
0answers
31 views

If $d(x_0,y_j)\to d(x_0,y_0)$, then $y_j \to y_0$.

Consider a metric space $X$, and a compact subset $C\subset X$.Let $x_0\in X-C$. We can show that there is a point $y_0\in C$ such that $d(x_0,y)=\inf_{y\in C} d(x_0,y)$. Now suppose there is ...
0
votes
1answer
24 views

generalize the question every every intersection of nested sequence of compact non-empty sets is compact and non-empty

I'm aware how to prove that the intersection of nested sequence of compact non-empty sets is compact and non-empty. but I want to generalize this question to transfer the hypothesis of having nested ...
3
votes
2answers
43 views

Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?

Let $X$ be a topological space space and $X=\cup_{i\in I} U_i$ a covering of $X$ by open subsets $U_i\subseteq X$. Is it true that $$ \operatorname{colim}\left(\coprod_{(i,j)\in I\times I} ...
1
vote
0answers
18 views

weak closure of unitary group in $B(H)$

Let $H$ be a Hilbert space with dim $H=\infty$ , and $\cal{U}$ be the group of all unitaries on $H$. Show that the weak closure of $\cal{U}$ is a semigroup with respect to the multiplication. I know ...
4
votes
2answers
65 views

Quotient Maps and Compact Hausdroff Spaces

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X$ and $Y$ are compact Hausdroff space and $f:X\rightarrow Y$ is a continuous ...
1
vote
1answer
25 views

What does continuity of a function mapping a topological space to a real line interval mean?

It makes sense for continuity to be defined on a function mapping a real line to a real line. Or how continuity is defined on a function between two topological spaces (every preimage of an open set ...
9
votes
3answers
244 views

Does every cover have an irredundant subcover?

While composing an answer for this question, I got troubled by a technical point. I wanted to assert the existence of an irredundant subcover of a given open cover, but realized I'm not sure how to ...
1
vote
2answers
92 views

Why is indiscrete topology unmetrizable?

For instance, the indiscrete topology for $X$ cannot arise from a metric when $X$ has more than one point. One way to see this is to note that the complement of a one-point set in a metric space is ...
2
votes
1answer
23 views

Condition on closures implies discreteness of topology.

I'm supposed to prove that if $(X, \tau)$ is a topological space such that $\overline{A \cap B} = \overline{A} \cap \overline{B}$ for all $A,B \subset X$, then $\tau$ is the discrete topology. My ...
2
votes
0answers
56 views

$\mathsf{Top}$ with proper maps has products.

In I.M. James' General Topology and Homotopy Theory, he presents proper maps before introducing compact sets, by defining $\phi:X \to Y$ to be proper iff $\phi \times \text{Id}_T$ is closed for all $T ...
2
votes
2answers
60 views

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
-1
votes
2answers
24 views

The image of a path-connected set under a continuous map is path-connected

Show that if $X$ is path-connected and $f:X\to Y$ is a continuous map, then the image $f(X)$ is path-connected. In order to show this is path connected I know the definition is :
1
vote
1answer
29 views

Show that the finite complement topology is connected

I am looking at $\mathbb{R}^n$ with the finite complement topology and need to show it's connected. I know that a connected doesn't have any non-trivial clopen sets. For $U \in T$ where T is the ...
-1
votes
1answer
25 views

Topology show X is compact

I no the following where we can use the definition of compact to be:
1
vote
1answer
27 views

A partition of the unit square such that the quotient space is the Klein bottle

Write down a partition $X^*$ of the unit square $X=[0,1]\times[0,1]$ such that the quotient space is the Klein bottle. I understand the definition of Quotient topology and Partitions, however, ...
1
vote
2answers
37 views

Topology without tears exercises 1.2 #6 i)

Let T be a topology on a set X such that T consists of precisely four sets; that is , $T = \{X, \emptyset, A, B\}$, where $A$ and $B$ are non empty distinct proper subsets of $X$. Prove that $A$ and ...
0
votes
0answers
29 views

Solution Verification: Prove in detail that the open rectangles in the Euclidean plane form an open base

I want some verification and/or some polishing on my proof. However if it is good, please let me know (I think this is highly unlikely to happen). Problem. Prove in detail that the open rectangles ...
4
votes
1answer
46 views

Extending a topology to linear combinations?

Suppose I have a topological space $X$, and some arbitrary field $K$. I am trying to nicely describe a set of functions on ${}_K X$, the set of $K$-linear combinations of values in $X$. I feel like ...
0
votes
1answer
14 views

Let $(X, \mathfrak T)$ be a topological space and supposed that $A$ is a subset of $X$ then the Bd(A) is a closed set.

Let $(X, \mathfrak T)$ be a topological space and supposed that $A$ is a subset of $X$ then the Bd(A) is a closed set. I am in an introduction to proofs class. I have to decided if this is a true ...
4
votes
2answers
46 views

How to show that there exists a sequence in $[0,1]$ such that the set of accumulation points of the sequence is $[0,1]$

This is related to homework but I am trying to find a special case first and see if I can generalize it. The problem is to construct some sequence $(x_n)$ in $[0,1]$ such that the accumulation points ...
1
vote
1answer
12 views

Let $A$ be a subset of $X$. Define $\mathfrak T = \{ U: A \subseteq U\} \cup \{\emptyset\}$. Then $\mathfrak T$ is a topology on $X$.

Let $A$ be a subset of $X$. Define $\mathfrak T = \{ U: A \subseteq U\} \cup \{\emptyset\}$. Then $\mathfrak T$ is a topology on $X$. I think this is a true statement and I therefore need to prove ...
1
vote
1answer
34 views

Why is this a convex polygon?

Let $\text{E}(2)$ be the group of isometries of the plane $\mathbb R^2$. Then $\text{E}(2)=\text{O}(2)\times\mathbb R^2$ as a topological space and is the semi-direct product as groups. Let $G$ be a ...
-1
votes
0answers
38 views

definition of largest circle containing in a convex body

For a convex body B(compact convex set with non-empty interior) what does each of these following values mean? 1- $$\ \sup_{x\in B}\inf_{y\in cB} d(x,y) $$ and 2- $$\ \sup_{x\in B}\inf_{y\in ...