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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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Interior Set of Rationals. Confused!

Can someone explain to me why the interior of rationals is empty? That is $\text{int}(\mathbb{Q}) = \emptyset$? The definition of an interior point is "A point $q$ is an interior point of $E$ if there exists a ball at $q$ such that the ball is…
Lemon
  • 12,664
14
votes
1 answer

Is $X/A$ contractible if both $X$ and $A$ are?

Let $X$ be a contractible topological space. Let $A\subseteq X$ be a contractible subspace. Is the quotient space $X/A$ necessarily contractible? It is not hard to show that this is true if, for example, the pair $(X,A)$ has the homotopy extension…
Mr. Frog
  • 698
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1 answer

Is there "essentially only 1" Jordan arc in the plane?

Let $p : [0,1] \to \mathbb{R}^2$ be continuous and injective. Does there always exists a continuous function $f : [0,1]\times \mathbb{R}^2 \to \mathbb{R}^2$ such that For all $x\in \mathbb{R}^2$, $f(0,x) = x$ and For all $t\in [0,1]$, $x\mapsto…
user57159
14
votes
1 answer

Neighborhood Base (the definition)

In Steven G. Krantz' A Guide To Topology, a countable neighborhood base is defined: Let $(X,U)$ be a topological space. We say that a point $x\in X$ has a countable neighborhood base at $x$ if there is a countable collection…
roo
  • 5,598
14
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3 answers

Can continuity be proven in terms of closed sets?

The usual definition of a continuous map between two topological spaces is that a map is continuous if the preimage of every open set is open. I believe, but am not sure, that to prove a map is continuous it suffices to show that the preimage of…
smackcrane
  • 3,211
13
votes
1 answer

On different definitions of neighbourhood.

I am going through the basics of topology, mainly to refresh them. I had taken a course some years ago but never used topology actively. So I am reading Munkres's Topology. I have noticed that he defines the notion of a neighbourhood in a way…
PaulRenet1
  • 271
13
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3 answers

homeomorphism of topological spaces is an equivalence relation ?

Would it be ok to say that homeomorphism of topological spaces is an equivalence relation ? I know that there isn't a base "set of all topological space" but since I encountered this phrase in several places, I believe it either can be made…
temo
  • 5,237
13
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2 answers

When is the union of topologies a topology?

The union of two topologies on some set may or may not be a topology. When is it a topology?
spin
  • 11,956
13
votes
2 answers

Smallest topology containing a family of other topologies on a set $X$

Let $T_a$ be a family of topologies on a set $X$. What is the smallest topology containing all the $T_a$? Obviously, the smallest it could be is the union of all the $T_a$, but that's not always a topology. So is it just the topology generated by…
A l'Maeaux
  • 816
13
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3 answers

How can I prove that this function is continuous at $0$?

The classical example of a functions with only one point of continuity is $$ f(x) = \begin{cases} x & \text{if } x \in \mathbf{Q}, \\ 0 & \text{otherwise}. \end{cases} $$ I only want to prove the continuity at $0$, using the definition in…
Daniel
  • 3,053
13
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3 answers

The closure of a product is the product of closures?

If $\{X_j:j\in J\}$ is a family of topological spaces and $A_j\subseteq X_j$, is it true that $\displaystyle\overline{\Pi_{j\in J}A_j}=\displaystyle{\Pi_{j\in J}\overline{A_j}}$? Is there an easy way to prove this? Of course, we are considering in…
Talexius
  • 2,015
13
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1 answer

Wrapped hexagon topology

When a square is wrapped at the edges, obtains a torus. Is it possible to do the same for a hexagon? What is the shape/topology of wrapping a hexagon? Note: I was just reading this…
whoplisp
  • 233
13
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2 answers

Prove: the set of zeros of a continuous function is closed.

Prove: the set of zeros of a continuous function is closed. And should the function on a closed interval?
HyperGroups
  • 1,393
13
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4 answers

The space obtained by identifying the antipodal points of a circle

Take the unit circle $S^1$ in $\Bbb{R}^2$ and partition it into subsets which contain exactly two points, the points being antipodal (at opposite ends of a diameter). $P$ is the resulting identification space. Now, what is homeomorphic to $P$? My…
Xena
  • 3,853
13
votes
3 answers

Accumulation points of accumulation points of accumulation points

Let $A'$ denote the set of accumulation points of $A$. Find a subset $A$ of $\Bbb R^2$ such that $A, A', A'', A'''$ are all distinct. I can find a set $A$ such that $A$ and $A'$ are distinct, but not one where $A,A',A'',A'''$ are all distinct.
John Michael
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