0
votes
1answer
32 views

Let C be a circle. Show that the only subset of C homeomorphic to a circle is C itself

I am trying to answer the question stated in the title. The hint in my book says to realize that for any z on the circle C{z} is still connected. I believe I can deal with case that shows that C{z} ...
1
vote
2answers
71 views

Why is this function continuous? (Topology)

Consider a space $Y = (\mathbb R,\mathcal F)$, where $U$ is element of $\mathcal F$ iff $U$ is empty or $U$ contains the number $1$. (It can be proven that $Y$ is topological space). Let $y_1,y_2$ be ...
1
vote
1answer
62 views

Using the intermediate value theorem for derivatives to infer that a function is strictly monotonic

My textbook Elementary Classical Analysis claims that by Darboux's theorem (the intermediate value theorem for derivatives), if a function $f:\mathbb R\to\mathbb R$ has a nonzero derivative on ...
2
votes
1answer
51 views

Function for which taking preimages preserves limit points

Suppose we have a surjection $f : X \to Y$ between topological spaces. What is the weakest assumption on $f$, $X$ and $Y$ you can think of that endows $f$ with the following property? If $A ...
2
votes
2answers
91 views

Homeomorphism is to topology as continuity is to

I have two questions. Are there examples where continuity does not preserve connectedness? Is there a structure whose structural properties are preserved by continuous map? (Just like homeomorphism ...
1
vote
1answer
219 views

Preimage of open pathwise connected set is pathwise connected

Is it a fact that under a continuous function, the preimage of an open, pathwise connected set is pathwise connected itself? I'm trying to prove that $GL_n(\mathbb{C})$ is pathwise connected, without ...
2
votes
2answers
212 views

Is a function $f:X\to Y$ continuous if and only if its graph on each connnected component of $X$ is connected?

I was thinking about this question today. Is the following true: Let $X$ be a topological space with connected components $\{C_i\}_{i\in I}$. Let $Y$ be a topological space and let $f:X\rightarrow ...
6
votes
2answers
293 views

Relationship between connectedness and continuity

Let $f:\mathbb R^n\to \mathbb R$. $f$ is continuous, The graph of $f$ if connected in $\mathbb R^{n+1}$ We define "connected" to be cannot be separated by 2 disjoint non-empty open set. My ...
2
votes
1answer
50 views

A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
4
votes
1answer
593 views

Why is the union of non disjoint path connected sets path connected?

If $X$ is a topological space, and $A, B\subset X$ are not disjoint and both path connected, it intuitively makes sense that $A\cup B$ is path connected: If, for $x_0\in A$ and $x_1\in B$ you take an ...
0
votes
1answer
53 views
6
votes
4answers
993 views

Is there a short proof for the Intermediate Value Theorem

My final for my introductory analysis course is tomorrow and my teacher gave us a list of possible theorems to prove. If anyone could please show me a proof for The Intermediate Value Theorem that is ...
8
votes
1answer
154 views

Path Connectedness and continuous bijections

Mathoverflow. Are there any two topological spaces $X$ and $Y$ such that they are path connected and such that there exist continuous bijections $X\rightarrow Y$ and $Y\rightarrow X$, but and yet ...
3
votes
2answers
317 views

If $f:X \rightarrow \mathbb{Q}$ is continuous, then $f$ must be constant

Let $X$ be a connected topological space. Prove if $f:X\to\mathbb Q$ is continuous, then $f$ must be constant. I know the definition of continuous is: for all $x\in X$ and all neighbourhoods $N$ of ...
0
votes
0answers
96 views

Continuous functions that preserve locally finite collections

Call a set locally finite component if its connected components form a locally finite collection. What conditions on a continuous function $f : X \to Y$ can guarantee that the image of any locally ...
4
votes
2answers
136 views

Continuity, Real Analysis

Some T/F questions. Instead of doing strict proofy questions, I am trying to understand the topic and making sure whether I am clear on the topic. Let me know whether I am right or wrong and I'll ...
5
votes
3answers
201 views

Continuity of function mapping connected set to connected set

If a function maps every connected set onto a connected set, is it necessarily continuous? I know the converse is true.
1
vote
1answer
80 views
8
votes
1answer
492 views

Is a continuous function simply a connected function?

Intuitively, a function $\mathbb{R}\rightarrow\mathbb{R}$ is continuous if you can draw its graph without taking the pen off the page. This suggests the following theorem: A map $f:X \rightarrow Y$ ...