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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.

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Describe the topology induced on the set $\mathbb N$ of positive integers by the euclidean topology on $\mathbb R$.

Describe the topology induced on the set $\mathbb N$ of positive integers by the euclidean topology on $\mathbb R$. Let $n \in \mathbb N$ then we know $(n - \frac{1}{2}, n + \frac{1}{2})$ is open in the euclidean topology on $\mathbb R$. Now, by…
user229672
  • 667
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X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to exactly one on your list.

I'm teaching my self topology with the aid of a book. I'm trying to do the following problem: Let X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to exactly one on your list. I'm not sure If I totally…
Michael Maliszesky
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Clarification on the Hausdorff property

If $X$ is a Hausdorff space then for points $a,b \in X$ there are disjoint open sets $U$ and $V$ such that $a \in U$ and $b \in V$. So, take a set of points $\{a_1, \ldots , a_n\}$ and another point $x$. Then for each $a_i$ there are disjoint open…
Alex Petzke
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The Euler characteristic of S1: a paradox

The Euler characteristic of $S^1$ should be equal to that of a triangle, the two being homeomorphic to each other. But it is $1$ for the triangle and zero for $S^1$; how does it make sense?
Arash
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Cantor Bendixson Theorem

Cantor Bendixson Theorem: Every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. This definition differs a bit from that in wikipedia. I have proved that 'If $X$ is a…
Katlus
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Show every irreducible subset of a topological space $X$ is contained in a maximal irreducible subset

Let $X$ be a topological space. A subset $A$ is irreducible if for every open $U,V\subseteq A$, we have $U\cap V\neq\varnothing$. Show that any irreducible subset $A\subseteq X$ is contained in a maximal irreducible set. So here's basically what I…
Alex Mathers
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open and closed sets in discrete space

I am confusing how to determine the set is clopen, neither open or closed, open but not closed and closed but not open. I read an example from "Topology without Tears". Let $X=\{a,b,c,d,e,f\}$ and…
Hing Yee
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Show that $\overline{U\cap \overline{A}}=\overline{U\cap A}.$

Show that: For every open set $U$ in a topological space $X$ and every $A\subset X$ we have $$\overline{U\cap \overline{A}}=\overline{U\cap A}.$$ The simple and new proof is welcome. Thanks for any help.
Paul
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Hausdorff spaces and separated maps.

Since the definitions for a Hausdorff space and for a separated map is so similar, I wondered about the implications between them. Suppose $X$ is a Hausdorff space, and we have a map $f:X \rightarrow Y$ to some topological space $Y$. So assuming…
Auclair
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The cone of a topological space is always path-connected

I'm having trouble proving that if $X$ is a topological space, then $C(X)$ (defined as $X \times [0,1] /\sim$ where $(x_1,t_1)\sim (x_2,t_2)$ iff they are equal or $t_1=0=t_2$) is path-connected. I understand the definition and I know I must take…
user178318
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A question on the compact subset

This is an exercise from a topological book. Let $X$ is Hausdorff and $K$ is a compact subset of $X$. $\{U_i:i=1,2,...,k\}$ is the open sets of $X$ which covers $K$. How to prove that there exist compact subsets of $X$: $\{K_i:i=1,2,...,k\}$ such…
Paul
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These two spaces are not homeomorphic...right?

why is $\Bbb R\times[0,1]\not \cong \Bbb R^2$? we can't use the popular argument of deleting a point and finding that one has more path components than the other here. So my idea is to delete a strip $\{0\}\times[0,1]$ from $\Bbb R\times[0,1]$. But…
Pindakaas1998
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Show that any line in $\Bbb R^2$ is closed

Show that any line in $\Bbb R^2$ is closed under the usual topology I think showing the complement is open is best way. Let $ax+by+c=0$ be a line in $\Bbb R^2$ as a line it is the set of $(x,y) \in \Bbb R^2 $ such that it satisfies the above…
oliverjones
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Is the notion of a proper map useful between non-locally-compact spaces?

Recall that a map $f: X \to Y$ between topological spaces is called proper if, for every compact $K \subseteq Y$, $f^{-1}(K)$ is compact. It strikes me that this definition is unlikely to be useful if $Y$ doesn't have "enough" compact subsets. And…
tcamps
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topology related to Binary space

I ran upon this topology based on binary space, perhaps using obscure terminology, but I am curious what it is and its properties. Let binary space be the set of strings of $0,1$'s, and let $S$ be the set of all functions that map the binary space…
verticese
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