Mathematics Stack Exchange
Mathematics Stack Exchange

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
19
votes
1 answer

The Number of Topologies on a Finite Set

I would like to know if there is like a magical formula to know how many topologies exist on a finite set For example for $X = \{ a, b, c \}$ I found $29$, but I dont know if there are more or how to know this exact number without writing all…
Gmath
  • 412
19
votes
1 answer

Continuity of a function to the integers

I am trying to prove that in $\mathbb{Z}$ with co-finite topology the only path-connected components are the singletons. (I reckon that) showing that "if a function $\gamma : [0,1] \to \mathbb{Z}$, where $\mathbb{Z}$ has co-finite topology, is…
user39280
19
votes
1 answer

Homogeneous topological spaces

Let $X$ be a topological space. Call $x,y\in X$ swappable if there is a homeomorphism $\phi\colon X\to X$ with $\phi(x)=y$. This defines an equivalence relation on $X$. One might call $X$ homogeneous if all pairs of points in $X$ are…
Rasmus
  • 18,404
18
votes
1 answer

Visualisation of the smash product

Wedge product, join, etc.,—all of them are no problem for my head, but I am really failing to get a visual idea of what the smash product wants to tell me. For example if I take two spheres, I have no idea what the smash product looks like. The…
user159356
18
votes
3 answers

Is a countable, totally disconnected Hausdorff space necessarily totally separated? How about zero-dimensional?

I'm starting to feel a little bad about using this website as my own personal counterexample generator, but here I go again... Terminology: Let's call a space zero-dimensional if it is $T_0$ and admits a basis of clopen sets. By a standard…
Mike F
  • 22,196
18
votes
1 answer

Connected and irreducible topological spaces

A topological space is called connected if any presentation of $X$ as $X = V_1 \uplus V_2$ by disjoint open subsets implies that one of them is trivial ($V_1 = X$ or $V_2 = X$). By taking complement one can replace the word "open" by "close". A…
LinAlgMan
  • 2,924
18
votes
2 answers

Find noncontinuous $F : \Bbb{R}^{\Bbb{R}} \to \Bbb{R}$ such that if $f_n \to f$ in $\Bbb{R}^{\Bbb{R}}$, then $F(f_n) \to F(f)$

I'm reading the book General Topology by S. Willard and I came across the following problem. We have to find an example of a noncontinuous function $F:\mathbb R^\mathbb R\to \mathbb R$ with the property that whenever $f_n\to f$ in $\mathbb R^\mathbb…
Dusan
  • 400
18
votes
2 answers

A sort of inverse question in topology

Given topological spaces $X$ and $Y$, we often consider the collection of continuous functions, $f: X \rightarrow Y$. My question is, given two sets $X$ and $Y$, and a sub-collection $\{g_{i}\}$ of the collection of all functions from $X$ to $Y$, do…
Dylan Yott
  • 6,999
18
votes
4 answers

What is the smallest cardinality of a Kuratowski 14-set?

A subset $A$ of a topological space $X$ is called a Kuratowski 14-set if exactly 14 different sets (including $A$) can be obtained from $A$ by alternately taking closures and complements. Are there any examples of 14-sets having finite cardinality?…
John
  • 4,305
18
votes
10 answers

Infinite class of closed sets whose union is not closed

Give an example of an infinite class of closed sets whose union is not closed.
Chalie Her
  • 459
17
votes
3 answers

If a square is colored red and blue, must there be either a red path connecting the top and bottom, or a blue path connecting the left and right?

In the board game Hex, players take turns coloring hexagons either red or blue. One player tries to connect the top and bottom edges of the board, colored red; the other tries to connect the left and right edges, colored blue. It is known that a…
Tanner Swett
  • 10,624
17
votes
7 answers

Why can the intersection of infinite open sets be closed?

I learned that the union of open sets is always open and the intersection of a finite set of open sets is open. However, the intersection of an infinite number of open sets can be closed. Apparently, the following example illustrates this. In $E^2$,…
David Faux
  • 3,425
17
votes
1 answer

If every compact set is closed, then is the space Hausdorff?

I know that in a Hausdorff space, every compact set is closed. However, is it true that if every compact set is closed, then the space is necessarily Hausdorff?
Cleaner
  • 267
17
votes
4 answers

A question about connected sets in $\mathbb{R}^2$

Let $C\subseteq\mathbb{R}^2$ be connected, open and have a bounded complement. Let $u\in C$ and $f:[0,1]\rightarrow \mathbb{R}^2$ be a continuous injective function such that $f(0)=u$. It is also given that $f([0,1))\subseteq C,f(1)\notin C$. Does…
Amr
  • 20,030
17
votes
3 answers

Infimum is a continuous function, compact set

Let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous map. Show that if $Y$ is compact then the function $g: X \rightarrow \mathbb{R}$ defined by $g(x) = \inf \{f(x,y): y \in Y\}$ is also continuous. No clue here. Can you please help?
user6495
  • 3,957
1 2 3
…
99 100