1
vote
0answers
23 views

About level curves of a continuous function in a real square, and connectivity

Assume f is a continuous function on the (unit) square in real plane. Name the edges N,S,E and W in the natural way. Assume f is >0 at W edge and <0 on E edge. Intuitively it is clear that there ...
3
votes
1answer
74 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.
6
votes
1answer
598 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 ...
3
votes
1answer
36 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 = \{ ...
2
votes
1answer
47 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
75 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
74 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
58 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
97 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
313 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
291 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
346 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
58 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
787 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
56 views

Let $ f:(X, d) \mapsto (Y,d) $ be an mapping such that $ Graph (f) $ is connected. [duplicate]

Where $ X $ is connected. Does it imply $ f $ to be continuous?
7
votes
4answers
1k 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
159 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
355 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
99 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
141 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
247 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
83 views
8
votes
1answer
576 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 $f:X \rightarrow Y$ ...