2
votes
1answer
31 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
2
votes
2answers
41 views

Counterexamples for $f(\overline{A}) = \overline{f(A)}$ and $\overline{f^{-1}(B)} = f^{-1}(\overline{B})$ in (non-)continuous mapping $f: X \to Y$

Let $f$ be a mapping. Prove that the following three statements are equivalent. $f$ is continuous; $\forall A \subseteq X: f(\overline{A}) \subset \overline{f(A)}$; $\forall B \subseteq ...
3
votes
1answer
72 views

Show that f is onto.

Let $X$ be a compact connected Hausdorff space and $f:X\rightarrow X$ a continuous open map. Show that f is onto.
1
vote
1answer
51 views

continuity extension of exponential $f(x)= a^x$

Consider tha exponential function $f(x) = a^x$, where $f: \mathbb{Q} \to \mathbb{R}$. My problem is to show that it has unique extension and how am I going to define this one? Also, I used a ...
0
votes
1answer
38 views

Is it true that $\textrm{supp}(f)\subseteq K$ implies $f|_{\partial K}=0$?

Maybe this will be an elementary question but I need to clarify this. Let $X$ be a metric space and let $f:X\longrightarrow \mathbb R$ continuous. Suppose $\textrm{supp}(f)\subseteq K$ where $K$ is ...
0
votes
0answers
28 views

Does continuity follow from linearity on all or only finite-dimensional vector spaces

I'm currently reading an introduction book on topology. While solving one of its exercises I came across something odd. The exercise is: Let $E$ and $F$ be normed spaces, let $T:E \to F$ be linear, ...
2
votes
1answer
36 views

Relationship between Continuity and Countability

This is a consequence of one of the problems in elementary real analysis that I am attempting to solve. I have this doubt. Suppose $f$ is a continuous map from the reals to the reals. If the set ...
1
vote
1answer
55 views

Prove $p_k\circ f$ continuous $\implies$ f is continuous

Let $X_1,\dots X_n$ topological space and $p_k:X_1\times\cdots X_n\to X_k$ the projection to the kth component. Let $Y$ be topological space and $f:Y\to X_1\times\cdots\times X_n$ function s.t ...
1
vote
1answer
43 views

For continuous functions, preimage of open set is open.

Let $f$ be a continuous function from a metric space $X$ into $Y$. If $V\subset Y$ and $V$ is open, then show that $f^{-1}(V)$ is open. The proofs I've seen of the fact that open sets have open ...
0
votes
2answers
83 views

Characterization of continuity in terms of preimages of open sets

1--8 Theorem. If $A\subset \mathbb R^n$, a function $f:A\to \mathbb R^m$ is continuous if and only if for every open set $U\subset \mathbb R^m$ there is some open set $V\subset \mathbb R^n$ such ...
1
vote
1answer
47 views

Continuity definition and theorem in a topology

This is an extremely common theorem, I have a function $f$ that maps $f:(X,\mathscr{S})\to(Y,\mathscr{T})$. I want to show that $f$ is continuous if and only if for all $V\in \mathscr{T}$, ...
2
votes
1answer
52 views

Topology, Proof of function being continuous

Let $ (X_i,d_i),(Y_i,d_i^*)$, $i=1,\ldots,n $ be metric spaces. Let $ f_i:X_i \to Y_i, i=1,...,n $ be continuous functions. Let $$ X = \prod_{i=1}^{n} X_i , Y = \prod_{i=1}^{n} Y_i $$ and ...
1
vote
2answers
55 views

Uniqueness of continuous extension from $A$ to $\overline{A}$ for maps into a Hausdorff space

I want to prove the following. Let $A$ be a subset of $X$. Let $f:A \to Y$ be continuous. Let $Y$ be Hausdorff. Show that if $f$ can be extended to a continuous function $g:\overline{A}\to Y$, ...
1
vote
0answers
43 views

Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
1
vote
0answers
24 views

On characterization of Riesz homomorphisms on $C(X)$ space

How to prove the following: Let $K$ be an arbitrary topological space and $\pi: C(K)\to\mathbb R$ be a map with $\pi (1) = 1$. If $\pi$ is a algebra homomorphism then it is an Riesz homomorphism.
1
vote
2answers
55 views

A question on the purpose of the condition on hausdorff to prove homeomorphism

This is a theorem proved in Munkres. Let $f:X\to Y$ be a bijective continous function. If X is compact and Y is hausdorff, then f is a homeomorphism. I knew Y being hausdorff which will be good to ...
2
votes
1answer
34 views

Extending a homeomorphism of the open disk to the boundary.

Let $D^2 = \{x \in \mathbb{R}^2 : ||x||\leq 1\}$ denote the closed disk and $int(D^2)$ denote its interior. If I have a homeomorphism $\ f: int(D^2) \rightarrow int(D^2)$ it is clear that it is not ...
1
vote
3answers
110 views

Exponetial map from real line to circle

Is the map $x\to e^{ix}$ from real line $\Bbb R$ to circle open? If I take any closed or half closed subset instead of $\Bbb R$ then this is definitely not open. But I'm little bit confused when ...
6
votes
1answer
596 views

Proof: X is connected

Just came from an exam and I am wondering how to prove the following: A topological space $X$ is connected if for each continuous function $f:X\rightarrow X$ there is a $x \in X$ such that ...
18
votes
2answers
430 views

A topological function with only removable discontinuities

I've posted similar questions here and here, but no one has answered them to my satisfaction. Suppose that $f:\mathbb{R} \to \mathbb{R}$ is such that $\lim_{y\to x}f(y)$ exists for all $x$, that is, ...
1
vote
2answers
55 views

Examples of continuous non-transitive group actions

In studying topology, I encountered this problem: Let $S$ be a topological space and let $G$ be a topological group acting continuously on $S$ (group action as $G \times S \to S$ map is continuous). ...
1
vote
3answers
99 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
0
votes
0answers
44 views

Prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$

I need to prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$. I define the map $h:\mathbb{R} \times S^1 \to \mathbb{R^2} \setminus \{(0,0)\}$ by ...
5
votes
3answers
71 views

Relation between continuous maps and convergence of sequences

I am studying metric spaces and I know that in a normed space $E$ a map $T:E \to E$ is contínuous if and only if $T(x_n) \to T(x)$ for every convergent sequence $x_n \to x$ in $E$. In my notes there ...
4
votes
3answers
42 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
5
votes
0answers
41 views

Properties of first-countable spaces

Hi I have questions regarding first-countable spaces. I just want to confirm something: The following are properties regarding limits and continuity of first countable spaces on Wikipedia: If $f$ ...
0
votes
1answer
27 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
1
vote
1answer
41 views

Intuition behind homeomorphism from $B((0, 0), 1) \to \mathbb{R^2}$

In my notes I have that the following function is a homemorphism from $B((0, 0), 1) \to \mathbb{R^2}$ $$h(x, y) \to \frac{f(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} (x, y)$$ where $f = ...
2
votes
1answer
33 views

Continous surjective map from $S^1$ to $S^n$

Is there any continous surjective map from $S^1$ or $[0,1]$ onto $S^n$, for some $n\geq 2$. Thank you.
0
votes
1answer
42 views

Weakest topology equivalence

Prove the equivalence of the following $Y \subset X$ has the subspace topology . $Y$ has the weakest topology to make the inclusion $i:Y\to X$ continuous. For all topological spaces $Z$ and maps ...
3
votes
0answers
38 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
1
vote
2answers
63 views

sequential continuity vs. continuity

A short and hopefully simple question for someone with more experience in topology: If a topology is induced by a mode of convergence and in fact nothing more is known apriori, wether if this topology ...
2
votes
1answer
41 views

Equivalence of continuous functions

Consider two topological spaces, $X$ and $Y$, and two continuous functions $f$ and $g$. By definition, given an open set $S$ in $Y$, the pre-image of $S$ under $f$ (or $g$) is an open set of $X$. Let ...
1
vote
2answers
87 views

on Continuous and Open Functions

Let $X,Y$ be compact Hausdorff spaces. If $f$ is a continuous function from $X$ onto $Y$, then $f$ is open. I am asking can the above result be proved. I am aware of the following cases: If $f$ ...
0
votes
1answer
84 views

Uniform Continuity $\implies$ Continuity

In metric spaces it is a well known fact that uniformly continuous functions are indeed continuous at any point. What about uniform spaces? How can I prove this? (with the definition of topology in ...
6
votes
1answer
85 views

Topology/continuous functions of $\mathbb{R}^n$ using paths.

We know that for a function $f:\mathbb{R}^n\to\mathbb{R}$ to be continuous, it is not sufficient for itto be continuous with respect to each coordinate. I believethe most commom counter-example is the ...
2
votes
1answer
60 views

Proving the metric attains a minimum on a compact subset

Let $(X,d)$ be a complete metric space. Suppose $B \subset X$ is compact. Prove that for every $a\in X$ the minimum $\min_{b\in B} d(a,b)$ exists. I'm pretty sure you can do this by just using the ...
1
vote
2answers
41 views

Is $f:\mathbb E^1\to X$ continuous?

$f(x)=x$. $X$ is the set of all real numbers with finite complement topology (A set is open in this space iff it's complement is finite).
3
votes
1answer
33 views

Show that $Y$ is not path-connected

Let $\mathbb{R}^2$ with the usual topology and let $$ Y = A_0 \cup (\bigcup_{n \in \mathbb{N}} A_n) \cup (\bigcup_{n \in \mathbb{N}}L_n)$$ where $$ A_0 = \{ 0 \} \times [0,1] \qquad A_n = \{ ...
5
votes
1answer
225 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 ...
0
votes
1answer
24 views

Tomae's Popcorn Function: Preimage of Opens?

I'm just wondering what the preimage of an (open) neighborhood say $(-0.5,0.5)$ containing the point $T(\frac{1}{\sqrt{2}})=0$ under Tomae's popcorn function $T$ looks like. Does somebody have an ...
0
votes
1answer
20 views

semicontinuity implies sequential semicontinuity

I have that $F:X\to (-\infty,+\infty]$, with $X$ topological space. By definition, $F$ is lower semicontinuous in $x_0 \in X$ if $\forall t \in \mathbb{R}: \. t<F(x_0) \.\exists U\in ...
1
vote
2answers
79 views

Continuous Map: Open $\iff$ Closed? [closed]

Is it true that a continuous map is open iff it is closed: $$f\text{ continuous}:\quad f\text{ open}\iff f\text{ closed}$$ The idea is that when somebody asks for embeddings, quotient maps and ...
0
votes
3answers
58 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
0
votes
1answer
39 views

Prove that any monotonic and bijective function is an homeomorphism with the usual topology

Let $f:(\mathbb{R},d_u\longrightarrow(\mathbb{R},d_u)$ an arbitrary function. Where $d_u$ is the usual distance. I have an exercise in which, with the sole asumptions of it being monotonic ...
0
votes
1answer
39 views

Proving that an arbitrary function is continuous (via topological methods)

Consider the following applications: $$f:(X,d)\longrightarrow(\mathbb{R^2},d_u)$$ $$\pi_i:(\mathbb{R^2},d_u)\longrightarrow(\mathbb{R},d_u)$$ Where $f$ is an arbitrary function, $\pi_i$ is the ...
0
votes
0answers
114 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
1
vote
3answers
111 views

Continuous map from the ring on the unit circle

Is there a surjective continuous map from the ring $r<x^2+y^2<1\,(0<r<1)$ on the unit circle $x^2+y^2<1$ ? It seems NO, but how can it be done ? Edit: what if we add the ...
2
votes
1answer
30 views

Relation between continuity and weak star continuity

Let us have a mapping $T:X^*\to X^*$. We can endow domain and codomain with norm 'strong' topology (let X be Banach space), or weak star topology. That gives us total 4 combinations: weak*-weak* ...
2
votes
1answer
91 views

Topological definition of continuity and its application to epsilon-delta definition?

So I am beginning Munkres' textbook on topology. The topological definition of continuity reads: $f:X\rightarrow Y$ is continuous if for each open subset $V\subset Y$, $f^{-1}(V)$ is an open subset ...