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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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 $x \times y$ ...
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Topological analog of the definition of continuity

The definition of continuity in topological spaces is given as: The function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at the point a in $\mathbb{R}^n $ iff given any open ball ...
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1answer
25 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 $ ...
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$\operatorname{Fr}( p(\overline U) )$ 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 ...
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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 ...
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1answer
10 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 ...
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68 views

Surgery to unlink $S^1$ and $S^2$ in $S^4$

Let us start with a $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ ...
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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 ...
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3answers
26 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 ...
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19 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 ...
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1answer
21 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 ...
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2answers
19 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 ...
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3answers
69 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 ...
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2
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51 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{ ...
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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 ...
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1answer
25 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
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16 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 ...
4
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1answer
44 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 ...
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10 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 ...
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3answers
241 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 ...
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2answers
50 views

$A$ and $B$ compact in a hausdorff space implies $A\cap B$ is compact [on hold]

Prove that if $A$ and $B$ are compact subset of a hausdorff space $X$, then $A$$\cap$$B$ is compact.
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1answer
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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 ...
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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 ...
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10 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
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1answer
290 views

two problem on homeomorphism and product of topological spaces

Let $\{(X_i, T_i) : i = 1,2,\ldots, n\}$ be a collection of topological spaces and let $\sigma$ be a permutation of the symbols $1, 2,\ldots, n$. For $i=1,2,\ldots, n$ let $Y_i = X_{ \sigma (i)}$ and ...
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36 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} ...
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1answer
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Homeomorphism between the 1-sphere and a semi-open real interval

I need help with a problem that's troubling me. In Lee's "Introduction to Topological Manifolds" I found this exercise: being given the exponential map $\ a:[0,1[\to\mathbb{S}^{1}$, $\ a(s)=e^{2\pi i ...
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2answers
37 views

Prove that exist bijection between inverse image of covering space

Let $B$ be path-connected and $p:E\to B$ covering map (with $E$ as covering space). Prove that $\forall a,b\in B$ exist 1-1 injection correspondence between $p^{-1}(a)$ and $p^{-1}(b)$ I thought ...
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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 ...
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29 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 ...
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0answers
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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 ...
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2answers
62 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 ...
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1answer
69 views

What is the motivation to continuous functions and measurable functions?

In topology the objects of interest are the space open sets, and a function will be continuous if the inverse image of any open set is an open set. In measure theory the objects of interest are the ...
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217 views

Are locally compact Hausdorff spaces with the homeomorphic one-point compactification necessarily homeomorphic themselves?

When practicing old qualifying exam problems, I had trouble with this one. Thanks for any help! Is it true that if the $1$-point compactifications of two locally compact Hausdorff spaces $X$, $Y$ are ...
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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 ...
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1answer
22 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 ...
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48 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 ...
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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 :
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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 ...
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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 ...
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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, ...
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1answer
25 views

Topology show X is compact

I no the following where we can use the definition of compact to be:
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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 ...
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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 ...
2
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1answer
32 views

Join of closed embeddings is a closed embedding

An exercise from James's book General Topology and Homotopy Theory asks the reader to prove that if $\phi_1:X_1 \to Y_1$ and $\phi_2:X_2 \to Y_2$ are closed topological embeddings, then $\phi_1 * ...
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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 ...
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1answer
45 views

Equivalent characterizations of ultrafilters

If $\mathcal{F}$ is a filter on $X$, will the below conditions be equivalent? (1) $\mathcal{F}$ is an ultrafilter. (2) For every $ \emptyset \neq M \subset X$, either $M \in \mathcal{F}$ or ...
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1answer
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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 ...