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Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

57719 questions
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For an n-gon, how many fundamental polygons are there?

I'm re-learning topological constructs and came across fundamental polygons as a way to describe certain topologies. I have an intuitive understanding but any mathematical understanding was left behind in a 20 year old college class. The examples…
Alex Feinman
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How is the Königsberg 7 bridge problem related to topology?

How can we use topology to solve the famous konigsberg 7 bridge problem? By using graph theory we can say that there does not exists any such path but I want to know the application of topology on the 7 bridge problem. Could anybody explain it to…
Prince Khan
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When does an Embedding extend into a Homeomorphism?

This is from a post in sci.math that did not get a full answer; I may repost it for the OP there: I am interested on the issue I read in another site of when an embedding from a closed set extends into a homeomorphism, i.e., if $C$ is closed in…
gary
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Is $[0,1)\times[0,1]$ a linear continuum?

I read that the topological space $X=[0,1)\times[0,1]$ with the dictionary order and order topology is not a linear continuum, as it does not satisfy the least upper bound property. (The definition of a linear continuum being a dense linear order…
Yong Pan
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Prove that a set is closed iff it contains all its accumulation points

Prove that a set is closed iff it contains all its accumulation points. I have no clue on how to approach the above problem. At first I would appreciate hints on how to get started in either direction $\Leftarrow$ or $\Rightarrow$. This is how we…
Olba12
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locally self-similar topologies

Call a topology "locally self-similar" if it has a basis in which each open set is homeomorphic to the entire space. What topologies have this property? So far, I have the following list: Any set with the indiscrete topology (the whole space is…
Hew Wolff
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Topology from interiors of closed sets?

Assume you've got an arbitrary topological space $X$. Now let $I$ be the set of the interiors of all closed subsets of $X$. And now assume you give me $I$, but don't tell me what $X$ is. Can I reconstruct the topology from $I$ alone? If it is not…
celtschk
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$A \subset \mathbb{R}$ such that $A$ is homeomorphic to $\mathbb{R} \setminus A$

For 2 topological spaces $(X,T_X)$ and $(Y,T_Y)$, I write $X \simeq Y$ if $X$ and $Y$ are homeomorphic. If $A \subset X$, I always endow $A$ with the subspace topology. I wonder if it is true that : $\exists A \subset \mathbb{R}$ such that $A…
Jeyrome Sapin
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Would this space be homeomorphic to the set of irrationals?

I've been reviewing various problems dealing with interesting homeomorphisms, and I came across this one. Is the product of the space of irrationals and the space of rationals homeomorphic to the space of irrationals? I haven't been able to make…
Maria
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Orbit space of torus homeomorphic to mobius strip

Let $\mathbb{T}^2$ be the torus and have $\mathbb{Z}_2$ action on the torus by permuting the coordinates. I am trying to prove the orbit space is congruent to the mobius strip $\mathbb{T}^2/\mathbb{Z}^2 \cong M$. I am writing $M\cong I^2/\alpha$,…
Nap D. Lover
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Why is the empty set in the standard topology?

The "standard topology" (put in quotations because I haven't verified that it is in fact a topology yet) is defined by $$\mathcal U \in \mathcal O_\text{standard} \iff \forall p\in \mathcal U \exists r\in \Bbb R^+ : B_r(p)\subseteq \mathcal U$$ Now…
user355875
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$x\in \bar{A}$ iff every neighbourhood of $x$ intersects $A$.

I read the proof of this theorem from Munkres, however I don't really understand intuitively why this is true. If someone could provide me of intuition of this theorem that would be nice. $\bar{A}$ is defined as the intersection of all closed sets…
user329017
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Help proving the primitive roots of unity are dense in the unit circle.

I'm having difficulty understanding how to prove that the primitive roots of unity are in fact dense on the unit circle. I have the following so far: The unit circle can be written $D=\{x\in\mathbb{C}:|x|=1\}$. The set of primitive $m$-th roots of…
pshmath0
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Local Path Connectedness and Cones

I am trying to prove the following: Let $X$ be a topological space. $CX$ is locally path connected if and only if $X$ is locally path connected, where $CX=X\times I/(X\times\{0\})$ I cannot seem to make much headway in either direction of the…
Holdsworth88
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what is the one point compactification of R?

It is said that the one point compactification of R is a circle. But how do i show it? I know it suffices to show R is homeomorphic to a punctured circle but how can i prove it?
Mathcho
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