# Tagged Questions

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### 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 ...
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### 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 = \{ ...
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### 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} ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f(x)\in V$ ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### Suppose $M$ is connected and suppose $f : M \rightarrow \mathbb R$ is continuous and only has irrational values, then $f$ is a constant function.

I want to do a proof by contradiction. You guys let me know if I goofing up.
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### 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 $f:X \rightarrow Y$ ...