Pointed space | Wikipedia audio article
Channel: wikipedia tts
Date: 31/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Pointed_space


00:01:24 Category of pointed spaces
00:02:41 Operations on pointed spaces



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.8914483026956876
Voice name: en-US-Wavenet-F


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics, a pointed space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x0, that remains unchanged during subsequent discussion, and is kept track of during all operations.
Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map f between a pointed space X with basepoint x0 and a pointed space Y with basepoint y0 is a based map if it is continuous with respect to the topologies of X and Y and if f(x0) = y0. This is usually denoted

f : (X, x0) → (Y, y0).Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint.
The pointed set concept is less important; it is anyway the case of a pointed discrete space.
Pointed spaces are often taken as a special case of the relative topology, where the subset is a single point.  Thus, much of homotopy theory is usually developed on pointed spaces, and then moved to relative topologies in algebraic topology.